Question

$AB$ is a variable line sliding between the coordinate axes in such a way that $A$ lies on x-axis and $B$ lies on y-axis. If $P$ is a variable point on $AB$ such that $PA=b,PB=a$ and $AB=a+b$, find the equation of the locus of $P$.

Answer 1

$A(l,0),B(0,h)$

$\therefore$ By section formula

$P(x,y)=(\frac{al}{a+b},\frac{bh}{a+b})\therefore x=\frac{al}{a+b},y=\frac{bh}{a+b}AB=\sqrt{l^2+h^2}=(a+b)\Rightarrow l^2+h^2={(a+b)}^2\therefore \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{l^2+h^2}{{(a+b)}^2}$

$\therefore \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ -required locus .