Question

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola `x^2/4 - y^2/2` = 1 at the point (x₁, y₁). Then `x_1^2 + 5y_1^2` is equal to ______.

Options

• 6

• 10

• 8

• 5

Answer 1

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola `x^2/4 - y^2/2` = 1 at the point (x₁, y₁). Then `x_1^2 + 5y_1^2` is equal to 6.

Explanation:

Line 2x – y = 0 is parallel to 2x – y = λ  

and is tangent to `x^2/4 - y^2/2` = 1  ...(i)

Tangent at (x₁, y₁)

`(x x_1)/4 - (yy_1)/2` = 1  ...(ii)

Equation (i) and Equation (ii) are same

∴ `(x_1/4)/2 = (y_1/2)/1 = 1/λ`

⇒ x₁ = `8/λ`, y₁ = `2/λ`

∴ This point satisfies hyperbola `(x_1^2)/4 - (y_1^2)/2` = 1

`64/(4λ^2) - 4/(2λ^2)` = 1

14 = λ²  ....(i)

Value of `x_1^2 + 5y_1^2 = 64/λ^2 + (5 xx 4)/λ^2`

= `84/λ^2`

= `84/14`  From equation (i)..

= 6