Introduction to Conic Section

A conic section is a section cut off from a right circular cone by a plane in various ways. The shape of the section depends upon the position of the cutting plane.

1. If the plane passes through the axis of the cone, the curve of intersection will be a pair of straight lines.

2. If the plane is perpendicular to the axis of the cone, the curve of intersection will be a circle.

3. If the plane is parallel to a generator of the cone (Say, PQ), the curve of intersection will be a parabola.

4. If the plane cuts the axis of the cone at an angle $\alpha$ (0 < $\alpha$ < $\pi$/2), the curve of intersection will be an ellipse.

5. If the plane is parallel to the axis of the cone, the curve of intersection will be a hyperbola.

DEFINITION OF CONIC

A conic section or conic is the locus of a point, which moves such that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line, not passing through the fixed point.

(i) The fixed point is called the focus.

(ii) The fixed straight line is called the directrix.

(iii) The constant ratio is called the eccentricity of the conic and is denoted by e.

(iv) When the eccentricity is unity; i.e., e = 1, the conic is called a parabola; when e < 1, the conic is called an ellipse; and when e > 1, the conic is called a hyperbola.

(v) The line of symmetry of the conic section is called its axis.

(vi) A point of intersection of a conic with its axis is called vertex.

(vii) The chord of a conic which passes through the focus and perpendicular to the axis is called the latus rectum.

GENERAL EQUATION OF CONIC

Let S ($\alpha$, $\beta$) be the focus and Ax + By + C = 0 be the equation of the directrix QN of the conic section.

Let P (x, y) be any point on it and let PN $\perp$ QN.

If 'e' be the eccentricity of the conic, then by definition,

$\frac{PS}{PN}$ = e $\Rightarrow$ PS² = e² . PN² ……(1)

$\Rightarrow$ (x – $\alpha$)² + (y – $\beta$)² = e² ${\left(\frac{Ax+By+C}{\sqrt{A^2+B^2}}\right)}^2$

which on simplification takes the form

ax² + 2hxy + by² + 2gx + 2fy + c = 0 ……(2)

where a, b, c, f, g and h are constants. (2) being the locus of P, is the equation of the conic.

RECOGNITION OF CONICS

The general equation of second degree ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents

Example -1: Find the equation of the parabola whose focus is (3, -4) and directrix is x - y + 5 = 0.

Solution: Let P(x, y) be any point on the parabola. Then

$\sqrt{{\left(x-3\right)}^2+{\left(y+4\right)}^2}=\frac{\left|x-y+5\right|}{\sqrt{1+1}}$ $\Rightarrow$ (x - 3)² + (y + 4)² = $\frac{{\left(x-y+5\right)}^2}{2}$

$\Rightarrow$ x² + y_­² + 2xy - 22x + 26y + 25 = 0 $\Rightarrow$ (x + y)² = 22x - 26y – 25.

Example -2: If (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then its equation is

(A) x² + 8y = 32 (B) y² + 8x = 32

(C) x² - 8y = 32 (D) y² - 8x = 32

Solution: AS = 2 = a. Vertex (0, 4) lies on y-axis. Hence the parabola X² = -4aY is a downward parabola as focus is below the vertex.

Or (x –0)² = -4 x 2(y –4)

Or x² + 8y = 32

Hence (A) is the correct answer.