The locus of the centroid of the triangle formed by any point P on the hyperbola 16x^2-9y^…

Question

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^2-9y^2+32x+36y-164=0$, and its foci is

Accepted Answer

$16(x+1)^2-9(y-2)^2=144\Rightarrow\dfrac{(x+1)^2}9-\dfrac{(y-2)^2}{16}=1$; $e=\dfrac53$, foci $(4,2)$ and $(-6,2)$.

Centroid $(h,k)$ of $P(\alpha,\beta)$ and the foci: $h=\dfrac{\alpha-2}3\Rightarrow\alpha=3h+2$; $k=\dfrac{\beta+4}3\Rightarrow\beta=3k-4$.

Substituting into the hyperbola: $16(3h+3)^2-9(3k-6)^2=144\Rightarrow16(h+1)^2-9(k-2)^2=16$, i.e.
$16x^2-9y^2+32x+36y-36=0.$

Solutions · 1