Question

$L_1$ is a tangent drawn to the curve $x^2-4y^2=16$ at $A(5,\frac{3}{2}).$ $L_2$ 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 $B$ respectively. Lines $L_1,L_2,L_3,L_4$ form 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)

Radius of the circle circumscribing the rectangle is $\frac{\sqrt{109}}{2}$

$\frac{x^2}{16}-\frac{y^2}{4}=1$
$(5,\frac{3}{2})$ lies on the curve.
Tangent at $A:$
$\frac{5x}{16}-\frac{3}{2}\left(\frac{y}{4}\right)=1$
$\Rightarrow L_1:5x-6y=16$


Tangent at $B:$
$L_2:5x-6y=-16$
Normal at $A:$
$6x+5y=\lambda$
Normal passes through $A(5,\frac{3}{2})$
$30+\frac{15}{2}=\lambda \Rightarrow \lambda =\frac{75}{2}$
$L_3:6x+5y=\frac{75}{2}$

Similarly, normal at $B$ is $L_4:6x+5y=-\frac{75}{2}$

Distance between tangents $2r=\frac{32}{\sqrt{25+36}}$
$\therefore r=\frac{16}{\sqrt{61}}$
Radius of circumcircle of rectangle is
$R=\sqrt{25+\frac{9}{4}}=\frac{\sqrt{109}}{2}$