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 $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$, whose one of the foci is the centre of $C$ and the length of…

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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 $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$, 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 ______

Aicharya · Accepted Answer

$r=\left|\frac{a+1}{\sqrt{2}}ight| \Rightarrow(a+1)^2=2 r^2$ Also $\left(\frac{3 a-1}{\sqrt{13}}ight)^2+\left(\frac{2}{\sqrt{13}}ight)^2=r^2$ $\begin{gathered} \Rightarrow\left(\frac{3 a-1}{\sqrt{13}}ight)^2+\frac{4}{13}=\frac{(a+1)^2}{2} \ 5 a^2-14 a-3=0 \end{gathered}$ $	herefore \quad a=-\frac{1}{5}, 3$ $\begin{aligned} & \because \quad a 
eq-\frac{1}{5} \Rightarrow a=3 \ & \Rightarrow \quad r=2 \sqrt{2} \end{aligned}$ $\because$ One focus of $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$ is $(3,0)$ $\begin{aligned} & \Rightarrow \alpha e=3 	ext { and } 2 \alpha=4 \sqrt{2} \ & \Rightarrow \alpha=2 \sqrt{2} \Rightarrow \alpha^2=8 \ & \alpha^2\left[1+\frac{\beta^2}{\alpha^2}ight]=9 \ & \alpha^2+\beta^2=9 \ & \Rightarrow \beta^2=1 \ & 	herefore 2 \alpha^2+3 \beta^2=19\end{aligned}$

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…

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