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.

Distance between the given points = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

Distance between A(1, -3) and P(x, y):

$= \sqrt{(x - 1)^2 + (y - (-3))^2}\\[1em] = \sqrt{(x - 1)^2 + (y + 3)^2}\\[1em] = \sqrt{x^2 + 1 - 2x + y^2 + 9 + 6y}\\[1em] = \sqrt{x^2 - 2x + y^2 + 10 + 6y}\\[1em]$

Distance between B(-2, 2) and P(x, y):

$= \sqrt{(x - (-2))^2 + (y - 2)^2}\\[1em] = \sqrt{(x + 2)^2 + (y - 2)^2}\\[1em] = \sqrt{x^2 + 4 + 4x + y^2 + 4 - 4y}\\[1em] = \sqrt{x^2 + 4x + y^2 + 8 - 4y}\\[1em]$

It is given that the distances of point P(x, y) from the points A(1, -3) and B(-2, 2) are in the ratio 2 : 3.

$⇒\dfrac{PA}{PB} = \dfrac{2}{3}\\[1em] ⇒\dfrac{\sqrt{x^2 - 2x + y^2 + 10 + 6y}}{\sqrt{x^2 + 4x + y^2 + 8 - 4y}} = \dfrac{2}{3}\\[1em] ⇒\dfrac{x^2 - 2x + y^2 + 10 + 6y}{x^2 + 4x + y^2 + 8 - 4y} = \dfrac{4}{9}\\[1em] ⇒9(x^2 - 2x + y^2 + 10 + 6y) = 4(x^2 + 4x + y^2 + 8 - 4y)\\[1em] ⇒9x^2 - 18x + 9y^2 + 90 + 54y = 4x^2 + 16x + 4y^2 + 32 - 16y\\[1em] ⇒9x^2 - 18x + 9y^2 + 90 + 54y - 4x^2 - 16x - 4y^2 - 32 + 16y = 0\\[1em] ⇒ 5x^2 + 5y^2 - 34x + 70y + 58 = 0\\[1em]$

Hence, proved :- 5x² + 5y² - 34x + 70y + 58 = 0.