Question

let $P$ and $Q$ are any two points on the circle $x^2+y^2=4$ such that $PQ$ is a diameter. If $L_1$ and $L_2$ are the lengths of perpendicular from $P$ and $Q$ on $x+y=1$, then the maximum value of $L_1,L_2$ is

• A. $1/2$
• B. $7/2$
• C. $1$
• D. $2$

Answer 1

Answer: (B)

$7/2$

Let one end point of the circle be $(2\cos t,2\sin t)$ so the diametric point to it would be $(2\cos (180+t),2\sin (180+t))$ that is $(-2\cos t,-2\sin t).$ The value of $L_1\times L_2$ would be -

$\Rightarrow \left[\frac{(2(\cos t+\sin t)-1)}{\sqrt{2}}\right]\left[\frac{2(\cos t+\sin t)+1}{\sqrt{2}}\right]$

$\Rightarrow \left[\frac{4{(\cos t+\sin t)}^2-1}{2}\right]$

now, maximum value $(\cos t+\sin t)$ is $\sqrt{2}$

maximum value of $L_1\times L_2=\frac{[4\left(2\right)-1]}{2}$

$=\frac{8-1}{2}$

$=3.5$ or $\frac{7}{2}$

Hence, the answer is $\frac{7}{2}.$