Question

P and Q are any two points on the X and Y axes: respectively such that PQ =7. If the point R divides PQ internally in the ratio $4:3$, find the equation of locus of R.

Answer 1

$|PQ|=7$

$\Rightarrow \sqrt{x^2+y^2}=7$

$\Rightarrow x^2+y^2=49$

$(\alpha ,\beta )$ divides $PQ$ in ratio $(4:3)$

$\therefore \alpha \frac{4x+3\left(0\right)}{(4+3)}$ and $\beta \frac{4\left(0\right)+3\left(y\right)}{(4+3)}$

$\alpha =\frac{4x}{7}$ and $\beta \frac{3y}{7}$

$\Rightarrow x=\frac{7\alpha }{7};y=\frac{7\beta }{3}$

but $x^2+y^2=49$ from (1)

$\therefore \left(\frac{7{\alpha }^2}{4}\right)+\left(\frac{7{\beta }^2}{3}\right)=49$

$\frac{49{\alpha }^2}{4}+\frac{49{\beta }^2}{3}=49$

$\Rightarrow \frac{{\alpha }^2}{4}+\frac{{\beta }^2}{3}=1$

$\therefore$ lows of $R$ is ab ellispse whose equation is $\frac{{\alpha }^2}{4}+\frac{{\beta }^2}{3}=1$