A point $P(x,y)$ moves in such a way that its distance from the point $A(3,1)$ is always three times its distance from the straight line $x=-1.$
My attempt is $${\sqrt {(x-3)^2 +(y-1)^2}} = 3{\sqrt {(x+1)^2 +(y-y)^2}}$$ The answer given is$$8x^2+9y^2-56x-18y+89=0$$ Is the answer wrong?
There is a general formula for conic section:
$$y^2=2px-(1-e^2)x^2$$
$p=a(1-e^2)$
Where $$e=\frac{distance from focus}{distance from .direct ix}$$
For ellipse $e<1$, for parabola $e=1$ and for hyperbola $e>1$ and $p=a(1-e^2)$ for ellipse and hyperbola.
In your question $e=3$ so the locus of points is hyperbola as your formula indicates too.