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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?

Jason Kim
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arthur anderson
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    The answer given describes a point whose distance from the line is three-times its distance from $A$. (Note that $(2,1)$ satisfies the equation in the "answer given". The distance from this point to the line is $3$, while its distance to $A$ is $1$.) So, the question and answer do not match. (Your approach to the question as stated is correct.) – Blue Sep 05 '18 at 14:13
  • @sirous: Actually, no. A parabola is the set of pts at the same distance from a given pt (a focus) and a given line (a directrix). If a conic's focus is closer than its directrix, then the conic is an ellipse; if the directrix is closer, a hyperbola. (One way to remember which is which is to think of a circle as the limiting case of an ellipse. The distance from the focus/center is some finite radius; the distance to the directrix is infinity. Clearly, the focus/center is closer to the points of the curve than the directrix.) Here, the focus is described as more-distant, so: hyperbola. – Blue Sep 06 '18 at 17:32

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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.

sirous
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