Let and be two fixed points on the -axis

MATHEMATICS · Multi-part · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Let

F_1(0,-c)

and

F_2(0,c)

be two fixed points on the

y

-axis. A point

P(x,y)

moves in the plane such that

|PF_2-PF_1|=2a

where

2a

is a constant. The curve traced by

P

has its centre at the origin and transverse axis along the

y

-axis. (i) Starting from the given condition

|PF_2-PF_1|=2a

, derive step by step the Cartesian equation of the curve in the form

\displaystyle \frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1

. (ii) Hence show that the parameters

a,\,b

and

c

satisfy the relation

c^{2}=a^{2}+b^{2}