JEE Advanced 2023 · Paper 2 · Mathematics Q13

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MathematicsInteger+4Question 13 of 51

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A = (4,1)$ , where $1 < r < 3$ . Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$ . The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$ . Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$ -axis at a point $B$ . If $AB = \sqrt5$ , then the value of $r^2$ is _____.

Your answer

Answer and worked solution

Official answer: 2

Key fact. The midpoint $M$ of a common tangent segment has equal tangent lengths to both circles: $MP=MQ$ . So $M$ lies on the radical axis.

Radical axis. $C_1:x^2+y^2-1=0$ and $C_2:(x-4)^2+(y-1)^2-r^2=0$ . Subtracting:
$8x+2y-18+r^2=0.$

Point B. The line through the two midpoints is this axis. It meets the $x$ -axis at
$B=\left(\frac{18-r^2}{8},0\right).$

Distance.
$AB^2=\left(4-\frac{18-r^2}{8}\right)^2+1=5\implies4-\frac{18-r^2}{8}=\pm2.$
This gives $r^2=2$ or $r^2=-30$ . Since $1<r<3$ , $r^2=\mathbf2$ .

Topics:Family of circles Tangents and normals Distance and section formulae

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JEE Advanced 2023 · Paper 2, Mathematics question 13