A point $P$ moves in such a way that the ratio of its distance from two coplaner points is always a fixed number $(\neq 1)$ . Then identify the locus of the point.

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Solution

Let two coplanar points $(0, 0) & (a, 0)$
distance between two point $= λ$
$\begin{matrix} \frac{\sqrt{{(x - 0)}^{2} + {(y - 0)}^{2}}}{\sqrt{{(x - a)}^{2} + y^{2}}} = λ \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{{(x - a)}^{2} + y^{2}}} = λ & [s q u a r i n g b o t h s i d e s] x^{2} + y^{2} + (\frac{λ^{2}}{λ^{2} - 1}) (a^{2} - 2 a x) = 0 \end{matrix}$
This is the equation of circle.

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A point P moves in such a way that the ratio of its distance from two coplaner points is always a fixed number (≠1). Then identify the locus of the point.

Let two coplanar points (0,0)&(a,0)distance between two point =λ√(x−0)2+(y−0)2√(x−a)2+y2=λ√x2+y2√(x−a)2+y2=λ[squaringbothsides]x2+y2+(λ2λ2−1)(a2−2ax)=0This is the equation of circle.

A point $P$ moves in such a way that the ratio of its distance from two coplaner points is always a fixed number $(\neq 1)$ . Then identify the locus of the point.