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Question: Let A and B be two fixed points and P, another point in the plane, moves in such a way that k<sub>1<...

Let A and B be two fixed points and P, another point in the plane, moves in such a way that k1PA + k2PB = k3, where k1, k2 and k3 are real constants. Then which one of the following is not the locus of P-

A

A circle, if k1 = 0 and k2, k3> 0

B

A circle, if k1> 0 and k2 < 0 and k3 = 0

C

An ellipse, if k1 = k2> 0 and k3> 0

D

An hyperbola, if k2 = –1 and k1, k3> 0

Answer

An hyperbola, if k2 = –1 and k1, k3> 0

Explanation

Solution

If k1 = 0, then k1PA + k2PB = k3

Ž PB = k3k2\frac{k_{3}}{k_{2}} > 0

Ž P describes a circle with B as centre and radius = k3k2\frac{k_{3}}{k_{2}}

If k3 = 0, then k1 PA + k2 PB = 0

Ž PAPB\frac{PA}{PB} = k2−k1\frac{k_{2}}{- k_{1}} = k > 0

Ž P described a circle with P1P2 as its diameter, P1 and P2 being the points which divide AB internally and externally in the ratio k : 1.

If k1 = k2 > 0 and k3 > 0, then

PA + PB = k3k1\frac{k_{3}}{k_{1}} = k > 0

Ž P describes an ellipse with A and B as its foci.