Question

Show that the set of all points such that the difference of their distances from (4, 0) and (– 4, 0) is always equal to 2 represent a hyperbola.

Answer 1

Let P(x, y) be any point.

We have `sqrt((x - 4)^2 + (y - 0)^2) - sqrt((x + 4)^2 + (y - 0)^2)` = 2

⇒ `sqrt(x^2 + 16 - 8x + y^2) - sqrt(x^2 + 16 + 8x + y^2)` = 2

Putting the x² + y² + 16 = z   ......(i)

⇒ `sqrt(z - 8x) - sqrt(z + 8x)` = 2

Squaring both sides, we get

⇒ `z - 8x + z + 8x - 2sqrt((z - 8x)(z + 8x))` = 4

⇒ `2z - 2sqrt(z^2 - 64x^2)` = 4

⇒ `z - sqrt(z^2 - 64x^2)` = 2

⇒ `(z - 2) = sqrt(z^2 - 64x^2)` = 2

⇒ `(z - 2) = sqrt(z^2 - 64x^2)`

Again squaring both sides, we have

z² + 4 – 4z = z² – 64x²

⇒ 4 – 4z + 64x² = 0

Putting the value of z, we have

⇒ 4 – 4(x² + y² + 16) + 64x² = 0

⇒ 4 – 4x² – 4y² – 64 + 64x² = 0

⇒ 60x² – 4y² – 60 = 0

⇒ 60x² – 4y² = 60

⇒ `(60x^2)/60 - (4y^2)/60` = 1

⇒ `x^2/1 - y^2/15` = 1

Which represent a hyperbola.

Hence proved.