Question

$L_1$ is a tangent drawn to the curve $x^2-4y^2=16$ at $A\left(5,\frac{3}{2}\right)\cdot L_2$ is another tangent parallel to $L_1$ which meets the curve at $B$. $L_3$ and $L_4$ are normals to the curve at $A$ and $B$. Lines $L_1,L_2,L_3,L_4$ forms a rectangle, then

• A. Equation of tangent at $B$ is $6y=5x-16$.
• B. Equation of normal at $B$ is $12x+10y+75=0$
• C. Radius of largest circle inscribed in the rectangle is $\frac{32}{\sqrt{61}}$.
• D. Radius of the circle circumscribing the rectangle is $\frac{\sqrt{109}}{2}$

Answer 1

Answer: (B), (D)

$\frac{x^2}{16}-\frac{y^2}{4}=1$
$\left(5,\frac{3}{2}\right)$ lies on the curve.
tangent at $A$
$\frac{5x}{16}-\frac{3}{2}\left(\frac{y}{4}\right)=1\Rightarrow 5x-6y=16$
tangent at B
$5x-6y=-6$
Normal at A
$6x+5y=\lambda$
$A\left(5,\frac{3}{2}\right)$
$30+\frac{15}{2}=1\Rightarrow \lambda =\frac{75}{2}$
$6x+5y=\frac{75}{2}$
$\Vert {|}^{ly}$ normal at $B$ is $6x+5y=\frac{-75}{2}$
distance between tangents is $2r=\frac{32}{\sqrt{25+36}}$
$\therefore r=\frac{16}{\sqrt{61}}$
Radius of circumcircle of rectangle is $R=\sqrt{25+\frac{9}{4}}=\frac{\sqrt{109}}{2}$