Find the points on the y-axis whose distance from the points (6,7) and (4,-3) are in the ratio 1 : 2.

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Solution

Let (x,y) be the req. point on y axis be (0,y)
given points $A (6, 7) B (4, - 3)$
$A = \sqrt{(0 - 6)^{2} + (y - 7)^{2}}$
$O B = \sqrt{(0 - 4)^{2} + (y + 3)^{2}}$
$\frac{O A}{O B} = \frac{1}{2}$
$2 O A = O B$
$4 O A^{2} = O B$
$4 [(6)^{2} + (y + 7)^{2}] = (4)^{2} + (y + 3)^{2}$
$4 [36 + y^{2} + 49 - 14 y] = 4^{2} + y^{2} + 9 + 6 y$
$y^{2} + 6 y + 25$
$4 y^{2} - 56 y + 340$
$3 y^{2} - 62 y + 315 = 0$
$3 y^{2} - 27 y - 35 y + 315 = 0$
$3 y (y - 9) - 35 (y - 9) = 0$
$(3 y - 35) (y - 9) = 0$
$y = \frac{35}{3} y = 9$
point $(0, 9) (0^{'}) (0, \frac{35}{3})$

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A point on the y-axis, whose distances from the points A(6,7) and B(4, -3) are in the ratio 1 : 2 is__________.

What are the points on y - axis whose distance from the line $\frac{x}{3} + \frac{y}{4} = 1$ is 4 units?

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