Question

Find the values of x, y if the distances of the point (x, y) from (-3, 0)  as well as from (3, 0) are 4.

Answer 1

We have P(x, y), Q(-3, 0) and R(3, 0)

`PQ = sqrt((x + 3)^2 + (y - 0)^2)`

`=> 4 = sqrt(x^2 + 9 + 6x + y^2)`

Squaring both sides

`=> (4)^2 = (sqrt(x^2 + 9 + 6x + y^2))`

`=> 16 = x^2 + 9 + 6x + y^2`

`=> x^2 + y^2 = 16 - 9 - 6x`

`=> x^2 + y^2 = 7 -  6x`   ......(1)

`PR = (sqrt((x - 3)^2 + (y - 0)^2)`

`=> 4 = sqrt(x^2 + 9 - 6x + y^2)`

Squaring both sides

`(4)^2 = (sqrt(x^2 + 9 - 6x + y^2))`

`=> 16 = x^2 + 9 - 6x + y^2`

`=> x^2 + y^2  = 16 - 9 + 6x`

`=> x^2 + y^2 = 7 + 6x`  .....(2)

Equating (1) and (2)

7 - 6x = 7 + 6x

⇒ 7 - 7 = 6x + 6x

⇒ 0 = 12x

⇒ x = 0

Substituting the value of x = 0 in (2)

`x^2 + y^2 = 7 + 6x`

`0 + y^2 = 7 + 6 xx 0`

`y^2 = 7`

`y = +- sqrt7`