Question

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

• A. Parabola
• B. Circle
• C. Hyperbola
• D. Pair of straight lines

Answer 1

The correct option is B

Circle

Explanation of the correct option:

Step 1. Let two coplanar points be $A(0,0)$ & $B(a,0)$

According to the question, $\frac{PA}{PB}=\lambda$

$\Rightarrow$$\frac{\surd {(x-0)}^2+{(y-0)}^2}{\surd {(x-a)}^2+y^2}=\lambda$

$\Rightarrow$ $\frac{\surd x^2+y^2}{\surd {(x-a)}^2+y^2}=\lambda$

Step 2. On squaring both sides, we get

$\Rightarrow$ $x^2+y^2={\lambda }^2{\left(x-a\right)}^2+y^2$

$\Rightarrow$$x^2+y^2+\left(\frac{{\lambda }^2}{{\lambda }^2-1}\right)(a^2-2ax)=0$

$\therefore$This locus of the point is a circle.

Hence, Option ‘B’ is Correct.