Question

Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.

Answer 1

Let (x, y) be any point.

Given points are (3, 0) and (9, 0)

We have `sqrt((x - 3)^2 + (y - 0)^2) + sqrt((x - 9)^2 + (y - 0)^2)` = 12

⇒ `sqrt(x^2 + 9 - 6x + y^2) + sqrt(x^2 + 81 - 18x + y^2)` = 12

Putting x² + 9 – 6x + y² = k

⇒ `sqrt(k) + sqrt(72 - 12x + k)` = 12

⇒ `sqrt(72 - 12x + k) = 12 - sqrt(k)`

Squaring both sides, we have

⇒ 72 – 12x + k = `144 + k - 24sqrt(k)`

⇒ `24sqrt(k)` = 144 – 72 + 12x

⇒ `24sqrt(k)` = 72 + 12x

⇒ `2sqrt(k)` = 6 + x

Again squaring both sides, we get

4k = 36 + x² + 12x

Putting the value of k, we have

4(x² + 9 – 6x + y²) = 36 + x² + 12x

⇒ 4x² + 36 – 24x + 4y² = 36 + x² + 12x

⇒ 3x² + 4y² – 36x = 0

Hence, the required equation is 3x² + 4y² – 36x = 0.