Question

Find the point on the X–axis which is equidistant from A(–3, 4) and B(1, –4).

Answer 1

Let point C be on the X-axis which is equidistant from points A and B.

Point C lies on X-axis.

∴ Its y co-ordinate is 0.

Let C = (x, 0)

C is equidistant from points A and B.

∴ AC = BC

According to distance formula,

`∴ sqrt([x  –  (– 3)]^2 + (0  –  4)^2] = sqrt((x  –  1)^2 + [0  –  ( –  4)]^2)`

∴ [x – (– 3)]² + (0 – 4)² = (x – 1)² + [0 – ( – 4)]²    ...(Squaring both the sides)

∴ (x + 3)²  + (– 4)² = (x – 1)² + 4²

∴ x² + 6x + 9 + 16 = x² – 2x + 1 + 16

∴ `cancelx^2 + 6x + 25 = cancelx^2  –  2x + 17`

∴ 6x + 2x = 17 – 25

∴ 8x = – 8

∴ `"x" = (-8)/8`

∴ x = –1

∴ The point on the X-axis, which is equidistant from points A and B, is (–1, 0).