Question

From a point $A$ on $x$-axis $2$ tangents are drawn to $x^2+y^2=16$ meeting $y$-axis at $P$ and $Q$, then

• A. minimum area of $\Delta APQ$ is $32$ sq units
• B. minimum value of $A{P}^2+A{Q}^2$ is $128$
• C. minimum value of $O{A}^2+O{P}^2$ is $64$
• D. For minimum area of $\Delta APQ$ the point $A$ is $(4\sqrt{2},0)$ or $(-4\sqrt{2},0)$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A minimum area of $\Delta APQ$ is $32$ sq units
B minimum value of $A{P}^2+A{Q}^2$ is $128$
C minimum value of $O{A}^2+O{P}^2$ is $64$
D For minimum area of $\Delta APQ$ the point $A$ is $(4\sqrt{2},0)$ or $(-4\sqrt{2},0)$

$OA=$ $4\sec \theta$
$OP=$ $4\text{cosec}\theta$

$O{A}^2+O{P}^2=16({\sec }^2\theta +{\text{cosec}}^2\theta )$

$=$ $16[2+{\tan }^2\theta +{\cot }^2\theta ]$

Minimum value of $O{A}^2+O{P}^2=64$

Now $A{P}^2+A{Q}^2=O{A}^2+O{P}^2+O{A}^2+O{Q}^2$
$=$ $2[O{A}^2+O{P}^2]$

Minimum value of $A{P}^2+A{Q}^2=128$

Also $\Delta APQ=\frac{1}{2}\left(PQ\right)\left(OA\right)=\frac{16}{\sin \theta \cos \theta }$

$=$ $\frac{32}{\sin 2\theta }\ge 32$

$\theta ={45}^{\circ },A(4\sqrt{2},0)$ or $(-4\sqrt{2},0)$