JEE Main 2025 (Online) 23rd January Morning ShiftQuestion 1
Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive $x$-axis, and cut off a chord of length $\dfrac{4}{\sqrt{13}}$ along the line $-3x+2y=1$. Let $H$ be the hyperbola $\dfrac{x^{2}}{\alpha^{2}}-\dfrac{y^{2}}{\beta^{2}}=1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2\alpha^{2}+3\beta^{2}$ is equal to $\_\_\_\_\_\_\_\_$ .
Hyperbola
Answer$19$
Solution
Centre $(h,0)$, $r=\frac{h+1}{\sqrt2}$; the chord condition $r^2-\frac{(3h+1)^2}{13}=\frac4{13}$ gives $5h^2-14h-3=0$, $h=3$, $r=2\sqrt2$. So $c=3$, $\alpha=2\sqrt2$, $\beta^2=9-8=1$ and $2\alpha^2+3\beta^2=19$.