Question

A$(4,0)$and B $(-4,0)$ are two given points.A variable point P is such that PA+PB=10 Show that the equation of locus of P is $\frac{x^2}{25}+\frac{y^2}{9}=1$

Answer 1

Given that:

$PA+PB=10$ means $2a=10$ or Semi major $a=5$,

Points $A(4,0)andB(-4,0)$ are the foci of the ellipse then

$AB=2ae$

$8=2ae$ [Since $AB=4-(-4)=8$]

$e=\frac{4}{5}.$

Now,

$b^2=a^2(1-e^2)=25(1-\frac{16}{25})=9$

Semi minor $b=3$,

On substituting the values of $a$ and $b$ in the general equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we get

$\frac{x^2}{25}+\frac{y^2}{9}=1$