Question

If $|z-4i|+|z+4i|=10$, then the equation of the locus of $z$ is:

• A. $\frac{x^2}{25}+\frac{y^2}{9}=1$
• B. $\frac{x^2}{5}+\frac{y^2}{9}=1$
• C. $\frac{x^2}{25}-\frac{y^2}{9}=1$
• D. $\frac{x^2}{9}+\frac{y^2}{25}=1$

Answer 1

The correct option is D $\frac{x^2}{9}+\frac{y^2}{25}=1$

Let $z=x+iy$
$\sqrt{x^2+(y-4{)}^2}+\sqrt{x^2+(y+4{)}^2}=10$
$\Rightarrow x^2+(y-4{)}^2=100+x^2+(y+4{)}^2-20\sqrt{x^2+(y+4{)}^2}$
$\Rightarrow 400(x^2+(y+4{)}^2)=(100+16y{)}^2$$=(100{)}^2+2\times 100\times 16y+256y^2$
$\Rightarrow 400x^2+400y^2+3200y+6400=(100{)}^2+3200y+256y^2$
$\Rightarrow 400x^2+144y^2=3600$
$\Rightarrow \frac{x^2}{9}+\frac{y^2}{25}=1$