Question

From the point $P(2,1)$,$a$ line of slope $m\in R$ is ran so as to cut the circle $x^2+y^2=1$ in points $A$ and $B$ if vary the slope $m$, the greatest possible value of $PA+PB$ will be

• A. $\frac{2}{\sqrt{5}}$
• B. $\frac{10}{\sqrt{5}}$
• C. $2\sqrt{5}$
• D. $4$

Answer 1

The correct option is C $2\sqrt{5}$

The greatest value of PA + PB can be when A and B are diametric points or in other words line through P passes through center of circle.

The general equation of circle is $x^2+y^2+2gx+2fy+c=0$ where,

Center = $(-g,-f)$

Radius = $\sqrt{g^2+f^2-c}$

Center of circle $x^2+y^2=1$ is (0,0). and radius is 1.

Since A and B are diametric points,

PA + PB

= PA + PA + AO + OB

= 2PO

$=2\sqrt{(2-0{)}^2+(1-0{)}^2}$

$=2\sqrt{5}$

Hence, (C) is the correct answer.