Let Q be a point on the circle B: x^2+y^2=a^2 | vedclass

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(D) Let $R(\alpha, \beta)$ be the required point,$Q(x_0, y_0)$ be a point on the circle,and $P(h, k)$ be a fixed point.Using the section formula,the c

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$\left(\frac{q h}{p+q}, \frac{q k}{p+q}\right)$

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(D) Let $R(\alpha, \beta)$ be the required point,$Q(x_0, y_0)$ be a point on the circle,and $P(h, k)$ be a fixed point.Using the section formula,the coordinates of $R$ are:$(\alpha, \beta) = \left(\frac{p x_0 + q h}{p+q}, \frac{p y_0 + q k}{p+q}\right)$From this,we get:$x_0 = \frac{(p+q)\alpha - qh}{p}$ and $y_0 = \frac{(p+q)\beta - qk}{p}$Since $Q(x_0, y_0)$ lies on the circle $x^2 + y^2 = a^2$,we substitute $x_0$ and $y_0$:$\left(\frac{(p+q)\alpha - qh}{p}\right)^2 + \left(\frac{(p+q)\beta - qk}{p}\right)^2 = a^2$Dividing by $\frac{(p+q)^2}{p^2}$,we get:$\left(\alpha - \frac{qh}{p+q}\right)^2 + \left(\beta - \frac{qk}{p+q}\right)^2 = \frac{p^2 a^2}{(p+q)^2}$Thus,the locus of $R(x, y)$ is a circle with centre $\left(\frac{qh}{p+q}, \frac{qk}{p+q}\right)$.

Let $Q$ be a point on the circle $B: x^2+y^2=a^2$ and $P(h, k)$ be a fixed point. If the locus of the point which divides the join of $P$ and $Q$ in the ratio $p: q$ is a circle $C$,then the centre of $C$ is

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