Question

$A$ is any point on $X-$axis and $B$ is any point on $Y-$axis and $p$ is a variable point on $AB$ with $PA=b$ and $PB=a$ and $AB=a+b$. Find the equation of locus of $P$.

Answer 1

point $P(h,k)$

$h=\frac{a{x}_0}{a+b},k=\frac{b{y}_o}{a+b}$

Form this $x_o=\frac{h(a+b)}{a},y_o=\frac{k(a+b)}{b}$

Also given that $AB=a+b=\sqrt{x_o^2+y_o^2}$

${\left(\frac{h(a+b)}{a}\right)}^2+{\left(\frac{k(a+b)}{b}\right)}^2=(a+b{)}^2$

S0 locus will be ${\left(\frac{x(a+b)}{a}\right)}^2+{\left(\frac{y(a+b)}{b}\right)}^2=(a+b{)}^2$