Question

The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x² + 5y² - 34x + 70y + 58 = 0.

Answer 1

It is given that PA: PB = 2: 3

`"PA"/"PB" = (2)/(3)`

`"PA"^2/"PB"^2 = (4)/(9)`

`((x - 1)^2 + (y + 3)^2)/((x + 2)^2 + (y - 2)^2) = (4)/(9)`

`(x^2 + 1 -2x + y^2 + 9 + 6y)/(x^2 + 4 + 4x + y^2 + 4 - 4y) = (4)/(9)`

9(x² - 2x + y² + 10 + 6y) = 4(x² + 4x + y² + 8 - 4y)

9x² - 18x + 9y² + 90 + 54y = 4x² + 16x + 4y² + 32 - 16y

5x² + 5y² - 34x + 70y + 58 = 0

Hence, proved.