Ellipse: Chord, Tangent and Normal

Q: When is a line y = mx + c tangent to an ellipse?

The line y = mx + c is tangent to x^2/a^2 + y^2/b^2 = 1 iff c^2 = a^2 m^2 + b^2. For every real m there are exactly two parallel tangents (± signs).

Q: What are the three forms of the tangent to an ellipse?

Slope form y = mx ± sqrt(a^2 m^2 + b^2); point form xx1/a^2 + yy1/b^2 = 1 at (x1, y1); parametric form x cos theta/a + y sin theta/b = 1 at (a cos theta, b sin theta).

Q: What is the director circle of an ellipse?

The director circle is the locus of points from which the pair of tangents to the ellipse are perpendicular. Its equation is x^2 + y^2 = a^2 + b^2, concentric with the ellipse.

Q: What are conjugate diameters of an ellipse?

Two diameters are conjugate if each bisects chords parallel to the other. Their slopes satisfy m1 m2 = -b^2/a^2 and the eccentric angles of their endpoints differ by pi/2. Also CP^2 + CD^2 = a^2 + b^2 and the parallelogram of tangents at their ends has area 4ab.

Q: What is the equation of a chord of an ellipse bisected at a given point?

If M(x1, y1) is the midpoint, chord equation is T = S1, i.e., xx1/a^2 + yy1/b^2 - 1 = x1^2/a^2 + y1^2/b^2 - 1.

Q: What is the reflection property of an ellipse?

A ray from one focus reflected off the ellipse passes through the other focus. The tangent at P bisects the external angle and the normal bisects the internal angle between the focal radii.

Q: What is the eccentricity condition for a focal chord in terms of eccentric angles?

If P(alpha) and P(beta) are the ends of a focal chord, then e = |cos((alpha - beta)/2) / cos((alpha + beta)/2)|. Equivalently tan(alpha/2) tan(beta/2) = (e-1)/(e+1) or (1-e)/(1+e).

Q: What is the difference between pole-polar and chord of contact?

For any point P(x1, y1), its polar with respect to the ellipse is xx1/a^2 + yy1/b^2 = 1. If P is outside, this polar is the chord of contact. If P is on the ellipse, the polar is the tangent at P. Pole-polar is the more general concept.

MathsEllipseFor JEE aspirants

This concept covers everything JEE Main and Advanced expect on lines, tangents, normals, and chords of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ . You will learn the tangency condition $c^{2} = a^{2} m^{2} + b^{2}$ , three forms each of tangent and normal, the pair-of-tangents rule $S S_{1} = T^{2}$ , the director circle $x^{2} + y^{2} = a^{2} + b^{2}$ , the chord of contact $T = 0$ , chord with a given midpoint $T = S_{1}$ , plus conjugate diameters, conormal (four normal) points, pole and polar, focal chord properties, and the reflection property that underpins whispering galleries. Each result is stated in the compact form used in JEE problems and illustrated with worked examples.

Key Formulas — Quick Reference

Line $y = m x + c$ meets the ellipse in real, coincident, imaginary points as $c^{2} <$ , $=$ , $> a^{2} m^{2} + b^{2}$ .
Tangency condition: $c^{2} = a^{2} m^{2} + b^{2}$ .
Tangent — slope form: $y = m x \pm a^{2} m^{2} + b^{2}$ .
Tangent — point form at $(x_{1}, y_{1})$ : $\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1$ .
Tangent — parametric form at $(a cos θ, b sin θ)$ : $\frac{x cos θ}{a} + \frac{y sin θ}{b} = 1$ .
Point of intersection of tangents at $α, β$ : $(\frac{a cos \frac{α + β}{2}}{cos \frac{α - β}{2}}, \frac{b sin \frac{α + β}{2}}{cos \frac{α - β}{2}})$ .
Normal — point form at $(x_{1}, y_{1})$ : $\frac{a ^{2} x}{x _{1}} - \frac{b ^{2} y}{y _{1}} = a^{2} - b^{2}$ .
Normal — parametric form: $a x sec θ - b y csc θ = a^{2} - b^{2}$ .
Normal — slope form: $y = m x - \frac{( a ^{2} - b ^{2} ) m}{a ^{2} + b ^{2} m ^{2}}$ .
Pair of tangents from $(x_{1}, y_{1})$ : $S S_{1} = T^{2}$ .
Director circle: $x^{2} + y^{2} = a^{2} + b^{2}$ .
Chord of contact from $(x_{1}, y_{1})$ : $T = 0$ , i.e., $\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1$ .
Chord with midpoint $(x_{1}, y_{1})$ : $T = S_{1}$ .
Eccentricity from focal chord $P (α), P (β)$ : $e = \frac{cos \frac{α - β}{2}}{cos \frac{α + β}{2}}$ .
Conjugate diameters: $C P^{2} + C D^{2} = a^{2} + b^{2}$ ; product of slopes $= - \frac{b ^{2}}{a ^{2}}$ .
Reflection: tangent bisects external angle, normal bisects internal angle between focal radii.

Throughout this concept we take the standard ellipse

S \equiv \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - 1 = 0

with

a > b

, and use the shorthand

S_{1} \equiv \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1

and

T \equiv \frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} - 1

1. Line and an Ellipse

Substitute $y = m x + c$ into $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ to get a quadratic in $x$ :

(a^{2} m^{2} + b^{2}) x^{2} + 2 a^{2} m c x + a^{2} (c^{2} - b^{2}) = 0.

The discriminant sign determines the intersection type:

Condition on $c^{2}$	Line meets the ellipse in
$c^{2} < a^{2} m^{2} + b^{2}$	two real distinct points (secant)
$c^{2} = a^{2} m^{2} + b^{2}$	two coincident points (tangent)
$c^{2} > a^{2} m^{2} + b^{2}$	imaginary points (line misses ellipse)

Hence: $y = m x + c$ is a tangent to the ellipse if and only if $c^{2} = a^{2} m^{2} + b^{2}$ .

Solved Example 1

Find the set of values of

λ

for which the line

3 x - 4 y + λ = 0

intersects the ellipse

\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1

at two distinct points.

Solution:

The line is $y = \frac{3 x + λ}{4}$ , so $m = \frac{3}{4}$ and $c = \frac{λ}{4}$ . Two distinct points require $c^{2} < a^{2} m^{2} + b^{2}$ :

\frac{λ ^{2}}{16} < 16 \cdot \frac{9}{16} + 9 = 18 ⟹ λ^{2} < 288 ⟹ - 122 < λ < 122 .

2. Tangents to an Ellipse

The tangent to $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ takes three standard forms, chosen according to what information is given:

2.1 Slope form

y = m x \pm a^{2} m^{2} + b^{2} .

For every real slope $m$ there are exactly two parallel tangents (the $\pm$ signs), touching the ellipse at the ends of a diameter.

2.2 Point form (at $(x_{1}, y_{1})$ on the ellipse)

\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1.

2.3 Parametric form (at $P (θ) = (a cos θ, b sin θ)$ )

\frac{x cos θ}{a} + \frac{y sin θ}{b} = 1.

Figure 1: Tangent and normal at

P (a cos θ, b sin θ)

on the ellipse. The tangent has slope

- \frac{b}{a} cot θ

; the normal is perpendicular to it at

P

and meets the major axis at

G

Point of intersection of tangents at $α$ and $β$ .

(a \frac{cos \frac{α + β}{2}}{cos \frac{α - β}{2}}, b \frac{sin \frac{α + β}{2}}{cos \frac{α - β}{2}}) .

Parallel tangents. The two tangents with the same slope

m

touch the ellipse at the ends of a diameter. The eccentric angles of their points of contact differ by

π

Solved Example 2

Find the equations of the tangents to the ellipse

3 x^{2} + 4 y^{2} = 12

which are perpendicular to the line

y + 2 x = 4

Solution:

The line has slope $- 2$ , so the tangent slope is $m = \frac{1}{2}$ . Rewrite the ellipse as $\frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1$ , so $a^{2} = 4$ , $b^{2} = 3$ .

Slope form: $y = \frac{1}{2} x \pm 4 \cdot \frac{1}{4} + 3 = \frac{1}{2} x \pm 2$ , i.e., $2 y = x \pm 4$ .

Solved Example 3

A tangent to the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

touches at a point

P

in the first quadrant and meets the coordinate axes at

A

and

B

. If

P

divides

A B

in the ratio

3 : 1

, find the equation of the tangent.

Solution:

Let $P \equiv (a cos θ, b sin θ)$ . The parametric tangent is $\frac{x cos θ}{a} + \frac{y sin θ}{b} = 1$ , meeting the axes at

A = (a sec θ, 0), B = (0, b csc θ) .

If $P$ divides $A B$ internally in $3 : 1$ (from $A$ to $B$ ), then

a cos θ = \frac{1 \cdot a sec θ + 3 \cdot 0}{4} = \frac{a sec θ}{4} ⟹ cos^{2} θ = \frac{1}{4}, cos θ = \frac{1}{2},

b sin θ = \frac{3 \cdot b csc θ}{4} ⟹ sin^{2} θ = \frac{3}{4}, sin θ = \frac{3}{2} .

Tangent: $\frac{x}{2 a} + \frac{3 y}{2 b} = 1$ , i.e., $b x + 3 a y = 2 ab$ .

Solved Example 4

Prove that the locus of the point of intersection of tangents to an ellipse at two points whose eccentric angles differ by a constant

α

is an ellipse.

Solution:

Let $P (h, k)$ be the intersection of tangents at $A (θ)$ and $B (β)$ . Using the formula,

h = \frac{a cos \frac{θ + β}{2}}{cos \frac{θ - β}{2}}, k = \frac{b sin \frac{θ + β}{2}}{cos \frac{θ - β}{2}} .

Then

(\frac{h}{a})^{2} + (\frac{k}{b})^{2} = sec^{2} \frac{θ - β}{2} .

Given $θ - β = α$ (constant), the RHS is constant $= sec^{2} (α /2)$ . Hence the locus is

\frac{x ^{2}}{a ^{2} sec ^{2} ( α /2 )} + \frac{y ^{2}}{b ^{2} sec ^{2} ( α /2 )} = 1,

which is an ellipse similar to the original.

3. Normals to an Ellipse

The normal at a point is perpendicular to the tangent there and passes through that point. Its three standard forms:

3.1 Point form (at $(x_{1}, y_{1})$ on the ellipse)

\frac{a ^{2} x}{x _{1}} - \frac{b ^{2} y}{y _{1}} = a^{2} - b^{2} .

3.2 Parametric form (at $P (θ) = (a cos θ, b sin θ)$ )

a x sec θ - b y csc θ = a^{2} - b^{2} .

3.3 Slope form

y = m x - \frac{( a ^{2} - b ^{2} ) m}{a ^{2} + b ^{2} m ^{2}} .

3.4 Four normals from an external point (conormal points)

From any point $P$ (not on the ellipse), at most four normals can be drawn to the ellipse. The four points on the ellipse at which these normals meet are called conormal points. If their eccentric angles are $θ_{1}, θ_{2}, θ_{3}, θ_{4}$ , then

θ_{1} + θ_{2} + θ_{3} + θ_{4} \equiv (2 n + 1) π (mod 2 π), n \in Z .

Additionally, the sum $\sum cos θ_{i} = 0$ and $\sum sin θ_{i} = 0$ (a consequence of Vieta's applied to the quartic in $tan (θ /2)$ ).

Solved Example 5

P

and

Q

are corresponding points on the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

and its auxiliary circle respectively. The normal at

P

to the ellipse meets

C Q

(extended) at

R

, where

C

is the centre. Prove that

C R = a + b

Solution:

Let $P \equiv (a cos θ, b sin θ)$ , so $Q \equiv (a cos θ, a sin θ)$ . Normal at $P$ :

(a sec θ) x - (b csc θ) y = a^{2} - b^{2} . ...(i)

Line $C Q$ : $y = tan θ \cdot x$ ... (ii).

Substitute (ii) into (i): $(a sec θ) x - (b csc θ) (tan θ) x = a^{2} - b^{2}$ , i.e., $(a - b) sec θ \cdot x = a^{2} - b^{2}$ , giving $x = (a + b) cos θ$ , and hence $y = (a + b) sin θ$ .

So $R \equiv ((a + b) cos θ, (a + b) sin θ)$ and $C R = x^{2} + y^{2} = a + b$ .

Solved Example 6

Find the shortest distance between the line

x + y = 10

and the ellipse

\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1

Solution:

Shortest distance between a curve and a line (non-intersecting) is measured along the common normal, so the tangent to the ellipse parallel to the line achieves the extremum.

Tangent parallel to $x + y = 10$ has slope $m = - 1$ . Slope-form tangent:

y = - x \pm 16 \cdot 1 + 9 = - x \pm 5.

Nearer tangent: $x + y - 5 = 0$ . Distance between parallel lines $x + y - 10 = 0$ and $x + y - 5 = 0$ :

distance = \frac{∣ - 10 - ( - 5 ) ∣}{1 ^{2} + 1 ^{2}} = \frac{5}{2} .

4. Pair of Tangents from an External Point

The combined equation of the two tangents that can be drawn from an external point $P (x_{1}, y_{1})$ to the ellipse is

S S_{1} = T^{2},

where

S \equiv \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - 1, S_{1} \equiv \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1, T \equiv \frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} - 1.

Figure 2: Pair of tangents from an external point

P (x_{1}, y_{1})

meets the ellipse at

A

and

B

. Combined pair equation:

S S_{1} = T^{2}

. The chord

A B

(chord of contact) has equation

T = 0

Solved Example 7

How many real tangents can be drawn from the point

(4, 3)

to the ellipse

\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1

? Find their equations and the angle between them.

Solution:

Check position: $S_{1} = \frac{16}{16} + \frac{9}{9} - 1 = 1 > 0$ , so $(4, 3)$ lies outside and two real tangents exist.

Pair of tangents $S S_{1} = T^{2}$ :

(\frac{x ^{2}}{16} + \frac{y ^{2}}{9} - 1) \cdot 1 = (\frac{4 x}{16} + \frac{3 y}{9} - 1)^{2} = (\frac{x}{4} + \frac{y}{3} - 1)^{2} .

Expanding and simplifying gives $- x y + 3 x + 4 y - 12 = 0$ , i.e., $(4 - x) (y - 3) = 0$ . So the two tangents are $x = 4$ and $y = 3$ , meeting at $(4, 3)$ at a right angle.

Angle between them $= \frac{π}{2}$ . (This is consistent with the director circle — see next section — since $(4, 3)$ satisfies $x^{2} + y^{2} = 25 = 16 + 9 = a^{2} + b^{2}$ .)

5. Director Circle

The locus of the point of intersection of two perpendicular tangents to the ellipse is a circle called the director circle:

x^{2} + y^{2} = a^{2} + b^{2} .

Its centre is the centre of the ellipse and its radius equals the length of the line joining the ends of the major and minor axes: $a^{2} + b^{2}$ .

Figure 3: The director circle

x^{2} + y^{2} = a^{2} + b^{2}

is the locus of points from which the two tangents to the ellipse are perpendicular to each other.

5.1 Quick derivation

Let $P (h, k)$ be the intersection of two perpendicular tangents. From the pair $S S_{1} = T^{2}$ , expand and gather coefficients of $x^{2}$ and $y^{2}$ . Perpendicularity gives (coefficient of $x^{2}$ ) + (coefficient of $y^{2}$ ) $= 0$ , which simplifies directly to $h^{2} + k^{2} = a^{2} + b^{2}$ .

Solved Example 8

An ellipse slides between two mutually perpendicular lines. Show that the locus of its centre is a circle.

Solution:

Let semi-axes be $a, b$ and centre $C (h, k)$ . The two perpendicular lines are tangents to the ellipse at every instant; their point of intersection lies on the director circle of the ellipse.

Take the perpendicular lines as the coordinate axes, so their point of intersection is the origin $O$ . Then $O C$ = radius of the director circle:

h^{2} + k^{2} = a^{2} + b^{2} ⟹ x^{2} + y^{2} = a^{2} + b^{2},

which is a circle of radius $a^{2} + b^{2}$ centred at the corner.

6. Chord of Contact

If two tangents from a point $P (x_{1}, y_{1})$ (outside the ellipse) touch the ellipse at $A$ and $B$ , then the line $A B$ is called the chord of contact of $P$ . Its equation is

T = 0, i.e., \frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1.

Solved Example 9

If tangents to the parabola

y^{2} = 4 a x

intersect the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

A

and

B

, find the locus of the point of intersection of the tangents to the ellipse at

A

and

B

Solution:

Let $P (h, k)$ be the intersection of tangents (to the ellipse) at $A$ and $B$ . Chord of contact of $P$ with respect to the ellipse:

\frac{h x}{a ^{2}} + \frac{k y}{b ^{2}} = 1. ...(i)

Line (i) is a tangent to the parabola $y^{2} = 4 a x$ , whose tangent in slope form is $y = m x + \frac{a}{m}$ , i.e., $m x - y = - \frac{a}{m}$ ... (ii).

Match coefficients of (i) and (ii):

\frac{m}{h / a ^{2}} = \frac{- 1}{k / b ^{2}} = \frac{- a / m}{1} ⟹ m = - \frac{h}{k} \cdot \frac{b ^{2}}{a ^{2}} and m = \frac{- ak}{b ^{2}} .

Eliminating $m$ : $- \frac{h b ^{2}}{k a ^{2}} = - \frac{ak}{b ^{2}}$ , giving $y^{2} = - \frac{b ^{4}}{a ^{3}} \cdot x$ .

7. Equation of a Chord with a Given Midpoint

The equation of the chord of the ellipse whose midpoint is $M (x_{1}, y_{1})$ is

T = S_{1}, i.e., \frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} - 1 = \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1.

Figure 4: Chord

A B

of the ellipse whose midpoint is

M (x_{1}, y_{1})

. Equation of such a chord is

T = S_{1}

Solved Example 10

Find the equation of the chord of the ellipse

\frac{x ^{2}}{36} + \frac{y ^{2}}{9} = 1

which is bisected at

(2, 1)

Solution:

Apply $T = S_{1}$ with $(x_{1}, y_{1}) = (2, 1)$ :

\frac{2 x}{36} + \frac{y}{9} - 1 = \frac{4}{36} + \frac{1}{9} - 1.

Simplify: $\frac{x}{18} + \frac{y}{9} = \frac{4}{36} + \frac{4}{36} = \frac{8}{36} = \frac{2}{9}$ , i.e., $\frac{x}{18} + \frac{y}{9} = \frac{2}{9}$ . Multiplying by $18$ : $x + 2 y = 4$ .

Solved Example 11

Find the length of the chord of the ellipse

\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1

whose midpoint is

(\frac{1}{2}, \frac{2}{5})

Solution:

By $T = S_{1}$ , the chord has slope

m = - \frac{b ^{2} x _{1}}{a ^{2} y _{1}} = - \frac{16 \cdot 1/2}{25 \cdot 2/5} = - \frac{8}{10} = - \frac{4}{5} .

So the chord is $y - \frac{2}{5} = - \frac{4}{5} (x - \frac{1}{2})$ , i.e., $4 x + 5 y = 4$ .

Substituting into the ellipse and simplifying leads to a quadratic in $x$ ; the distance formula gives length $= \frac{7 41}{5}$ .

8. Diameter of an Ellipse

The locus of midpoints of a system of parallel chords of an ellipse is a straight line passing through the centre, called a diameter. If the parallel chords have slope $m$ , the diameter that bisects them has equation

y = - \frac{b ^{2}}{a ^{2} m} x .

Every diameter of an ellipse passes through its centre and is bounded by the curve at two points diametrically opposite each other.

9. Conjugate Diameters

Two diameters of an ellipse are conjugate if each of them bisects all chords parallel to the other. Let the two diameters have slopes $m_{1}$ and $m_{2}$ respectively. Then $m_{1}$ and $m_{2}$ are conjugate iff

m_{1} m_{2} = - \frac{b ^{2}}{a ^{2}} .

Figure 5: Conjugate diameters

P P^{'}

and

D D^{'}

: the eccentric angles at

P

and

D

differ by

\frac{π}{2}

. Each diameter bisects all chords parallel to the other, and the tangents at the four ends enclose a parallelogram of area

4 ab

9.1 Eccentric-angle characterisation

If $P (θ)$ is one end of a diameter, then the ends of the conjugate diameter have eccentric angles $θ \pm \frac{π}{2}$ . Writing $C P$ and $C D$ for the semi-conjugate diameters,

C P^{2} + C D^{2} = a^{2} + b^{2} .

9.2 Parallelogram of tangents at ends of conjugate diameters

The four tangents at the ends of a pair of conjugate diameters $P P^{'}$ and $D D^{'}$ form a parallelogram whose area is $4 ab$ , a constant independent of which conjugate pair is chosen.

9.3 Equi-conjugate diameters

Two conjugate diameters are called equi-conjugate when $C P = C D$ . This happens when $θ = \pm \frac{π}{4}$ , giving $C P = C D = \frac{a ^{2} + b ^{2}}{2}$ . Their slopes are $\pm \frac{b}{a}$ , so the equi-conjugate diameters are along the asymptotes of the auxiliary rectangle.

Solved Example 12

Find the angle between two diameters of the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

whose extremities have eccentric angles

α

and

β = α + \frac{π}{2}

Solution:

Slopes: $m_{1} = \frac{b sin α}{a cos α} = \frac{b}{a} tan α$ and $m_{2} = \frac{b sin β}{a cos β} = - \frac{b}{a} cot α$ (using $β = α + π /2$ ).

Then

tan ϑ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}} = \frac{\frac{b}{a} ( tan α + cot α )}{1 - \frac{b ^{2}}{a ^{2}}} = \frac{2 ab}{( a ^{2} - b ^{2} ) sin 2 α} .

(Note that these two diameters are indeed a conjugate pair, since $m_{1} m_{2} = - b^{2} / a^{2}$ .)

10. Pole and Polar

For a fixed point $P (x_{1}, y_{1})$ (called the pole), the line

\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1

is called the polar of $P$ with respect to the ellipse. In terms of $T$ , the polar of $P$ is simply $T = 0$ .

10.1 Geometric meaning

If $P$ is outside the ellipse, its polar is the chord of contact of tangents from $P$ .
If $P$ is on the ellipse, its polar is the tangent at $P$ .
If $P$ is inside the ellipse, the polar is the locus of poles of all chords passing through $P$ .

10.2 Conjugate points

Two points $P$ and $Q$ are conjugate with respect to the ellipse if each lies on the polar of the other. If $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ , this is equivalent to

\frac{x _{1} x _{2}}{a ^{2}} + \frac{y _{1} y _{2}}{b ^{2}} = 1.

11. Focal Chord and Eccentricity

If $P (α)$ and $P (β)$ are the ends of a focal chord (chord passing through a focus $(\pm a e, 0)$ ), then the equation of the chord is

\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2} .

Substituting $(x, y) = (\pm a e, 0)$ gives $\pm e cos \frac{α + β}{2} = cos \frac{α - β}{2}$ , hence

e = \frac{cos \frac{α - β}{2}}{cos \frac{α + β}{2}} .

Applying componendo-dividendo (with the appropriate sign) yields the equivalent form:

tan \frac{α}{2} tan \frac{β}{2} = \frac{e - 1}{e + 1} or \frac{1 - e}{1 + e} .

Solved Example 13

Show that the locus of the point of intersection of the tangents at the extremities of any focal chord of an ellipse is the corresponding directrix.

Solution:

Let the chord join $P (α)$ and $P (β)$ and pass through the focus $(a e, 0)$ . The intersection of tangents at these two points is

(\frac{a cos \frac{α + β}{2}}{cos \frac{α - β}{2}}, \frac{b sin \frac{α + β}{2}}{cos \frac{α - β}{2}}) .

Using the focal-chord relation $e cos \frac{α + β}{2} = cos \frac{α - β}{2}$ (taking the " $+$ " sign for focus $(a e, 0)$ ), the $x$ -coordinate becomes

x = \frac{a cos \frac{α + β}{2}}{e cos \frac{α + β}{2}} = \frac{a}{e} .

So the locus is $x = \frac{a}{e}$ , the directrix corresponding to focus $S (a e, 0)$ .

12. Important Highlights (Reflection & Focal Geometry)

Figure 6: Reflection property: a ray from one focus

S^{'}

strikes the ellipse at

P

and reflects along

P S

through the other focus. The two marked angles are equal: the normal

P G

bisects the internal angle between the focal radii, and the tangent bisects the external angle.

Sum of focal distances. For any point $P$ on the ellipse with foci $S, S^{'}$ , we have $S P + S^{'} P = 2 a$ .
Reflection property. The tangent and normal at a point $P$ on the ellipse bisect the external and internal angles respectively between the focal distances of $P$ . Equivalently, a ray from one focus is reflected off the ellipse toward the other focus. As a corollary, the straight lines joining each focus to the foot of perpendicular from the other focus on the tangent at $P$ meet on the normal $P G$ and bisect it, where $G$ is the point where the normal meets the major axis.
Perpendicular from focus onto tangent. The product of lengths of the perpendicular segments from the two foci on any tangent is $b^{2}$ . The feet of these perpendiculars lie on the auxiliary circle. The tangents at these feet (to the auxiliary circle) intersect on the ordinate of $P$ , and the locus of their point of intersection is a similar ellipse.
Portion of tangent between point of contact and directrix. The segment of a tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
Normal at $P$ and its intercepts. Let the normal at $P$ meet the major axis at $G$ , the minor axis at $g$ , and let $C F$ be the perpendicular from the centre $C$ on this normal. Then
- (i) $P F \cdot P G = b^{2}$
- (ii) $P F \cdot P g = a^{2}$
- (iii) $P G \cdot P g = S P \cdot S^{'} P$
- (iv) $C G \cdot C T = C S^{2}$ , where $T$ is the point where the tangent at $P$ meets the major axis
- (v) The locus of the midpoint of $G g$ is another ellipse having the same eccentricity as the original.
Focal distance as diameter. The circle described on any focal distance as diameter touches the auxiliary circle. Perpendiculars from the centre upon all chords that join the ends of any pair of perpendicular diameters of the ellipse are of constant length.
Tangent intercept. If the tangent at $P$ on a standard ellipse meets the coordinate axes at $T$ and $t$ , and $C Y$ is the perpendicular from the centre on it, then
- (i) $T t \cdot P Y = a^{2} - b^{2}$ ,
- (ii) the least value of $T t$ is $a + b$ .

Common Mistakes to Avoid

Watch out

The pair-of-tangents relation $S S_{1} = T^{2}$ works only if the point $P (x_{1}, y_{1})$ lies outside the ellipse ( $S_{1} > 0$ ). If $S_{1} \leq 0$ , no real pair of tangents exists.
The chord of contact $T = 0$ and the chord with a given midpoint $T = S_{1}$ look similar; they are not the same equation. Use $T = 0$ when $P$ is outside and you want the join of the two tangent-touch points; use $T = S_{1}$ when $P$ is an interior midpoint of an existing chord.
The tangent slope-form $y = m x \pm a^{2} m^{2} + b^{2}$ is only valid for finite $m$ . For vertical tangents ( $x = \pm a$ ), use the point form.
The director circle exists only when $a^{2} + b^{2} > 0$ , which is always true for an ellipse. But for degenerate cases (e.g., a hyperbola with $a = b$ ), the "director circle" collapses to a point — do not confuse the two conics' director loci.
When applying $c^{2} = a^{2} m^{2} + b^{2}$ to test tangency, ensure $a$ corresponds to the coefficient of $x^{2}$ in the standard ellipse form $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ (denominator, not numerator). A common slip is to use the coefficient itself instead of its reciprocal.
Conjugate diameters are not the same as perpendicular diameters (except for a circle). Perpendicular diameters have $m_{1} m_{2} = - 1$ ; conjugate diameters have $m_{1} m_{2} = - \frac{b ^{2}}{a ^{2}}$ .
The number of normals from an external point is at most four, not always four. Some points admit only two real normals (with the other two complex). Do not force four real feet without checking.

Frequently Asked Questions

Q1. When is a line $y = m x + c$ tangent to an ellipse?

The line $y = m x + c$ is tangent to $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ if and only if $c^{2} = a^{2} m^{2} + b^{2}$ . This condition comes from setting the discriminant of the substituted quadratic in $x$ to zero. For every real $m$ , there are exactly two parallel tangents (the $\pm$ signs).

Q2. What are the three forms of the tangent to an ellipse?

Slope form: $y = m x \pm a^{2} m^{2} + b^{2}$ ; use when the slope is known. Point form: $\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1$ at $(x_{1}, y_{1})$ on the ellipse. Parametric form: $\frac{x cos θ}{a} + \frac{y sin θ}{b} = 1$ at $(a cos θ, b sin θ)$ ; use when the eccentric angle is known.

Q3. What is the director circle of an ellipse?

The director circle is the locus of points from which the two tangents to the ellipse are perpendicular. Its equation is $x^{2} + y^{2} = a^{2} + b^{2}$ : a circle concentric with the ellipse with radius $a^{2} + b^{2}$ , equal to the length of the diagonal of the rectangle formed by the major and minor axes.

Q4. How many normals can be drawn from a point to an ellipse?

At most four normals can be drawn from any point to an ellipse. The four points of contact are called conormal points, and their eccentric angles satisfy $θ_{1} + θ_{2} + θ_{3} + θ_{4} \equiv (2 n + 1) π (mod 2 π)$ . For some external points fewer than four normals may be real; the others are complex.

Q5. What are conjugate diameters of an ellipse?

Two diameters are conjugate if each bisects all chords parallel to the other. Their slopes satisfy $m_{1} m_{2} = - \frac{b ^{2}}{a ^{2}}$ . The eccentric angles of their endpoints differ by $\frac{π}{2}$ , and $C P^{2} + C D^{2} = a^{2} + b^{2}$ where $C P, C D$ are semi-conjugate diameters. The four tangents at their ends form a parallelogram of constant area $4 ab$ .

Q6. What is the equation of a chord of an ellipse bisected at a given point?

If $M (x_{1}, y_{1})$ is the midpoint of a chord, its equation is $T = S_{1}$ , i.e., $\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} - 1 = \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1$ . This is different from the chord of contact $T = 0$ , which is used when tangents are drawn from an external point.

Q7. What is the reflection property of an ellipse?

A ray of light emitted from one focus and reflected off the ellipse passes through the other focus. Equivalently, the tangent at any point $P$ bisects the external angle between the two focal radii $S P$ and $S^{'} P$ , and the normal bisects the internal angle. This is why whispering galleries have elliptical ceilings and lithotripsy devices use ellipsoidal reflectors.

Q8. What is the eccentricity condition for a focal chord in terms of eccentric angles?

If $P (α)$ and $P (β)$ are the endpoints of a focal chord, then $e = \frac{cos \frac{α - β}{2}}{cos \frac{α + β}{2}}$ . Equivalently, $tan \frac{α}{2} tan \frac{β}{2}$ equals $\frac{e - 1}{e + 1}$ or $\frac{1 - e}{1 + e}$ depending on which focus the chord passes through.

Q9. What is the difference between pole-polar and chord of contact?

For any point $P (x_{1}, y_{1})$ , its polar with respect to the ellipse is the line $\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1$ , and $P$ is called the pole of that line. When $P$ lies outside the ellipse, this polar coincides with the chord of contact of tangents from $P$ . When $P$ lies on the ellipse, the polar is the tangent at $P$ . Pole-polar is the more general concept.

Previous year questions on Ellipse: Chord, Tangent and Normal

14 questions from past papers, each with a step-by-step solution.

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1. Line and an Ellipse

2. Tangents to an Ellipse

2.1 Slope form

2.2 Point form (at (x1​,y1​) on the ellipse)

2.3 Parametric form (at P(θ)=(acosθ,bsinθ))

3. Normals to an Ellipse

3.1 Point form (at (x1​,y1​) on the ellipse)

3.2 Parametric form (at P(θ)=(acosθ,bsinθ))

3.3 Slope form

3.4 Four normals from an external point (conormal points)

4. Pair of Tangents from an External Point

5. Director Circle

5.1 Quick derivation

6. Chord of Contact

7. Equation of a Chord with a Given Midpoint

8. Diameter of an Ellipse

9. Conjugate Diameters

9.1 Eccentric-angle characterisation

9.2 Parallelogram of tangents at ends of conjugate diameters

9.3 Equi-conjugate diameters

10. Pole and Polar

10.1 Geometric meaning

10.2 Conjugate points

11. Focal Chord and Eccentricity

12. Important Highlights (Reflection & Focal Geometry)

Common Mistakes to Avoid

Frequently Asked Questions

Q1. When is a line y=mx+c tangent to an ellipse?

Q2. What are the three forms of the tangent to an ellipse?

Q3. What is the director circle of an ellipse?

Q4. How many normals can be drawn from a point to an ellipse?

Q5. What are conjugate diameters of an ellipse?

Q6. What is the equation of a chord of an ellipse bisected at a given point?

Q7. What is the reflection property of an ellipse?

Q8. What is the eccentricity condition for a focal chord in terms of eccentric angles?

Q9. What is the difference between pole-polar and chord of contact?

Previous year questions on Ellipse: Chord, Tangent and Normal

Ready to master Ellipse?

2.2 Point form (at $(x_{1}, y_{1})$ on the ellipse)

2.3 Parametric form (at $P (θ) = (a cos θ, b sin θ)$ )

3.1 Point form (at $(x_{1}, y_{1})$ on the ellipse)

3.2 Parametric form (at $P (θ) = (a cos θ, b sin θ)$ )

Q1. When is a line $y = m x + c$ tangent to an ellipse?