Question

$P$ is a point on the circle $C_1:q^2(x^2+y^2)=a^2{p}^2$

$Q$ is a point on the circle $C_2:x^2+y^2=a^2$

If the coordinates of $P$ are $(h,k)$ then the locus of the point which divides the join of $PQ$ in the ratio $p:q$ is a circle $C_3,$ whose centre is at the point

• A. $(\frac{hp}{p+q},\frac{kq}{p+q})$
• B. $(\frac{h}{p+q},\frac{k}{p+q})$
• C. $(\frac{hq}{p+q},\frac{kq}{p+q})$
• D. $(\frac{hp}{p+q},\frac{kp}{p+q})$

Answer 1

Answer: (C)

$(\frac{hq}{p+q},\frac{kq}{p+q})$

Let $Q\left(x_1{y}_1\right)$ be a point on $C_2.$

Let $(m,n)$ be the point which divides $PQ$ is the ratio $p:q,$ then
$m=\frac{hq+x_{1}p}{p+q}$ and $n=\frac{kq+y_{1}p}{p+q}$
$\Rightarrow x_1=\frac{m(p+q)-hq}{p}$ and $y_1=\frac{n(p+q)-kq}{p}$
Since $Q\left(x_1{y}_1\right)$ lies on $x^2+y^2=a^2$, we have

$\left(m\right(p+q)-hq{)}^2+(n(p+q)-kq{)}^2=a^2{p}^2$
$\Rightarrow {(m-\frac{hq}{p+q})}^2+{(n-\frac{kq}{p+q})}^2=\frac{a^2{p}^2}{(p+q{)}^2}$
Locus of $(m,n)$ is ${(x-\frac{hq}{p+q})}^2+{(y-\frac{kq}{p+q})}^2=\frac{a^2{p}^2}{(p+q{)}^2}$
Which is circle $C_3$ if $(\alpha ,\beta )$ denote the centre of $C_3$, then
$\alpha =\frac{hq}{p+q}$ and $\beta =\frac{kq}{p+q}$