Find the point on the y-axis whose distances fro the points A (6,7) and B(4,-3) are in the ratio 1:2 - Maths - Coordinate Geometry

Question

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

Lovina Kansal · Accepted Answer

Let the coordinates of the required point on y axis be (0, y).

Then

$\frac{\sqrt{{(0 - 6)}^{2} + (y - 7)^{2}}}{\sqrt{{(- 4)}^{2} + (y - (- 3))^{2}}} = \frac{1}{2} \Rightarrow \frac{6^{2} + (y - 7)^{2}}{{(- 4)}^{2} + (y + 3)^{2}} = \frac{1}{4}$

⇒ 4 (36 + y² + 49 – 14y) = 16 + y² + 9 + 6y

⇒ 4y² – 56y + 340 = y² + 6y + 25

⇒ 4y² – y² – 56y – 6y + 340 – 25 = 0

⇒ 3y² – 62y + 315 = 0

⇒ (3y – 35) (y – 9) = 0

$\Rightarrow y = \frac{35}{3}, 9$

Thus, the required points are $(0, \frac{35}{3}) and (0, 9)$
Note:

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