JEE Main 2026 Apr 6 Shift 2, Mathematics Q10: Circle: Chord, Tangent And Normal

JEE Main2026Apr 6, Shift 2Mathematics

Let $C$ be a circle having centre in the first quadrant and touching the $x$ -axis at a distance 3 units from the origin. If the circle $C$ has an intercept of length $63$ on $y$ -axis, then the length of the chord of the circle $C$ on the line $x - y = 3$ is:

A
8
B
6
C
$62$
D
$82$

Solution

Find the circle. The circle touches the $x$ -axis at $(3, 0)$ , so the centre is at $(3, k)$ with radius $k$ ( $k > 0$ ).

Length of $y$ -axis chord: $2 r^{2} - d^{2}$ where $d = 3$ (distance from centre to $y$ -axis):

$2 k^{2} - 9 = 63$ $\Rightarrow$ $k^{2} - 9 = 27$ $\Rightarrow$ $k = 6$ .

Centre $(3, 6)$ , radius $6$ .

Chord length on $x - y = 3$ . Perpendicular distance from $(3, 6)$ to the line:

$p = \frac{∣3 - 6 - 3∣}{2} = \frac{6}{2}$ .

Chord length $= 2 r^{2} - p^{2} = 2 36 - 18 = 218 = 62$ .

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JEE Main 2026 Apr 6 Shift 2, Mathematics Q10: Circle: Chord, Tangent And Normal

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