Question

If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P is

• A. A parabola
• B. An ellipse
• C. A hyperbola
• D. None of these

Answer 1

Answer: (B)

An ellipse

Answer 2

$\because \mathrm { PA } + \mathrm { PB } = 4$

⇒ $\left( \frac { h f - b g } { a b - h ^ { 2 } } , \frac { g h - a f } { a b - h ^ { 2 } } \right)$.(1)

Let $( x + a ) ^ { 2 } + y ^ { 2 } = l$ & $( x - a ) ^ { 2 } + y ^ { 2 } = m$

from (1)

$\sqrt { l } + \sqrt { m } = 4$ .........(2)

$( \sqrt { l } + \sqrt { m } ) ( \sqrt { l } - \sqrt { m } ) = 4 a x$ ......(3)

$\sqrt { l } - \sqrt { m } = a x$ ………….(4) (from (2))

adding (2) & (4) $\therefore 2 \sqrt { l } = ( 4 + a x )$

⇒ $4 l = 16 + a ^ { 2 } x ^ { 2 } + 8 a x$

⇒ $4 \left\{ x ^ { 2 } + y ^ { 2 } + 2 a x + a ^ { 2 } \right] = 16 + a ^ { 2 } x ^ { 2 } + 8 a x$⇒ $\left( 4 - a ^ { 2 } \right) x ^ { 2 } + 4 y ^ { 2 } = 16 - 4 a ^ { 2 }$ ellipse.