Question

If P(x₁, y₁) is a point on the hyperbola x² - y² = a², then SP. S'P = ______.

विकल्प

• `(x_1^2 - y_1^2)/"a"^2`

• `(x_1^2 + y_1^2)/"a"^2`

• `x_1^2 - y_1^2`

• `x_1^2 + y_1^2`

Answer 1

If P(x₁, y₁) is a point on the hyperbola x² - y² = a², then SP. S'P = `underline(x_1^2 + y_1^2)`.

Explanation:

Given, equation of hyperbola

x² - y² = a²

If P(x₁, y₁) is a point on Eq. (i), then

`x_1^2 + y_1^2 = "a"^2`

Now, SP = ex₁ - a

and SP' = ex₁ + a

∴ SP · SP' = (ex₁ - a)(ex₁ + a)

`= "e"^2 * x_1^2 - "a"^2`

`= 2x_1^2 - "a"^2   ...("since", "for"  x^2 - y^2 = "a"^2, "e" = sqrt2)`

`= 2x_1^2 - (x_1^2 - y_1^2)`    ....[using Eq. (i)]

`= x_1^2 + y_1^2`