Question

Answer the following:

Find the equations of the tangents to the hyperbola 3x² − y² = 48 which are perpendicular to the line x + 2y − 7 = 0

Answer 1

The equations of tangents to the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 in terms of slope m are

y = `"m"x ± sqrt("a"^2"m"^2 - "b"^2)`   ...(1)

The equation of the hyperbola is 3x² – y² = 48

i.e., `x^2/16 - y^2/48` = 1

Comparing this with `x^2/"a"^2 - y^2/"b"^2` = 1, we get,

a² = 16, b² = 48

Slope of x + 2y – 7 = 0 is `-1/2`

The required tangent is perpendicular to it

∴ its slope = m = 2

∴ by (1), the required equations of tangents are

y = `2x ± sqrt(16(4) - 48)`

∴ y = 2x ± 4