Question

$C_1$ and $C_2$ are two concentric circles having centre at the origin with radii $2\text{and}4,$ respectively. If $PA$ and $PB$ are two tangents to $C_1$ from a point $P$ that lies on $C_2$, then the locus of centroid of $\Delta PAB$ is

• A. $x^2+y^2=8$
• B. $x^2+y^2=4$
• C. $x^2+y^2=9$
• D. $x^2+y^2=1$

Answer 1

The correct option is B $x^2+y^2=4$



$C_1:x^2+y^2=4$
$C_2:x^2+y^2=16$

Let $P(h,k)$ be a point on circle $C_2$.
Then, $h^2+k^2=16$

The equation of the chord of contact of tangents drawn from a point $P(h,k)$ to $C_1$ is $hx+ky=4$

Let $A(x_1,y_1)$ and $B(x_2,y_2)$ be the points, where tangents touches $C_1$.
As $A$ and $B$ lie on both $C_1$ and chord of contact, the abscissa and the ordinates of $A$ and $B$ are roots of equations, $x^2+{\left(\frac{4-hx}{k}\right)}^2=4$
and ${\left(\frac{4-ky}{h}\right)}^2+y^2=4$ respectively.

i.e., $x_1,x_2$ are the roots of the equation $(h^2+k^2)x^2-8hx+4(4-k^2)=0$
and $y_1,y_2$ are the roots of the equation $(h^2+k^2)y^2-8ky+4(4-h^2)=0$

$\Rightarrow x_1+x_2=\frac{8h}{h^2+k^2}$
and $y_1+y_2=\frac{8k}{h^2+k^2}$

As $P(h,k)$ lies on $x^2+y^2=16$, we get
$x_1+x_2=\frac{h}{2}$ and $y_1+y_2=\frac{k}{2}$

Let $(\alpha ,\beta )$ be coordinates of the centroid of $\Delta PAB$.
Then, $\frac{x_1+x_2+h}{3}=\alpha$
and $\frac{y_1+y_2+k}{3}=\beta$
$\Rightarrow \frac{3h}{2\times 3}=\alpha$ and $\frac{3k}{2\times 3}=\beta$
$\Rightarrow h=2\alpha$ and $k=2\beta$
$\therefore 4{\alpha }^2+4{\beta }^2=16$

Hence, the locus is $x^2+y^2=4$