Question

Given base of a triangle and sum of its sides. Find the locus of vertex of triangle (which does not lie on base).

Answer 1

$BC\to$ base is given which means length of $BC$ is fixed.

Let $BC=k$

$A\to$ variable vertex

Let $S=AB+BC+AC$

$\Rightarrow AB+AC=S-K=$ constant

$\Rightarrow$ Sum of distances of variable point from two fixed points is constant and is greater than distance between $2$ point

$\Rightarrow$ This is physical definition of ellipse

$P{S}_1+P{S}_2=2k$

$\Rightarrow$ Locus of vertex $A$ is an ellipse.