Question

A pair of tangents $AC$ and $BD$ of lengths $a$ and $b$ respectively are drawn at the extremities of the diameter $AB$ of length $d$. $BC$ and $AD$ intersect each other at point $P$. If $a$, $d$ and $b$ are in G.P, then which of the following is/are correct?

• A. $P(\equiv ̸A,\equiv ̸B)$ lies on the circumference of the given circle
• B. $P$ lies on the director circle of the given circle
• C. The maximum distance of $P$ from $AB$ is $d$
• D. The maximum area of $\Delta ABP$ is $\frac{d^2}{4}$ sq. units

Answer 1

Answer: (A), (D)

The correct options are
A $P(\equiv ̸A,\equiv ̸B)$ lies on the circumference of the given circle
D The maximum area of $\Delta ABP$ is $\frac{d^2}{4}$ sq. units


$\tan \theta =\frac{b}{d},\tan \varphi =\frac{a}{d}$
$\tan \theta \tan \varphi =\frac{ab}{d^2}$
$a,d\text{and}b\text{are in G.P}$
$\Rightarrow d=\sqrt{ab}\Rightarrow \frac{ab}{d^2}=1$
$\therefore \tan \theta \tan \varphi =1$
$\Rightarrow \tan \theta =\cot \varphi$
$\Rightarrow \tan \theta =\tan ({90}^{\circ }-\varphi )$
$\Rightarrow \theta +\varphi ={90}^{\circ }$
$\therefore \angle APB={180}^{\circ }-(\theta +\varphi )={90}^{\circ }$
Since, diameter subtends a right angle at point $P$, therefore $P$ lies on the circumference of the circle.

​​​​​​Area of $\Delta ABP$ is maximum when its altitude become the radius of the circle i.e., $\frac{d}{2}$
$\Rightarrow \text{Maximum area}=\frac{1}{2}\times d\times \frac{d}{2}=\frac{d^2}{4}\text{sq. units}$