Find the point on y-axis which is equidistant from the points (5, -2) and (-3, 2).

Or

The line segment joining the points A(2, 1) and B(5, -8) is trisected at the points P and Q such that P is nearer to A. If P lies on the line given by 2x – y + k = 0, find the values of k.

Let, the equidistant point is C(0, y)     (on y axis, x coordinate is 0)

If it is equidistant from (5, -2) and (-3, 2) then AC = BC

We will get

&

As, AC = BC

Squaring both sides we get,
⇒ y² + 4y + 29 = y² - 4y + 13

⇒ 8y + 16 = 0

⇒ y = -2

So, the equidistant point is (0, -2)

OR

Consider the points A(2, 1) and B(5, -8) .

If P is one of the trisected points then the ratio of AB at P = 1: 2

Let, coordinates of P is (x, y).
By section formula which states that if a point P(x,y) divides the line with endpoints A(x₁,y₁) and B(x₂,y₂) in the ratio m:n, the coordinates of x and y are:

The coordinates of P are:



x = 3 and y = -2
So, the coordinates of P is (3, -2).

P satisfies the equation of the given line 2x – y + k = 0.

⇒ 2(3) – (-2) + k = 0

⇒ k = - 8