Question

Find the equations of the tangents to the hyperbola 5x² – 4y² = 20 which are parallel to the line 3x + 2y + 12 = 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)

Given hyperbola is 5x² – 4y² = 20

i.e. `x^2/4 - y^2/5` = 1

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

a² = 4, b² = 5

Slope of 3x + 2y + 12 = 0 is `-3/2`

The required tangent is parallel to it

∴ its slope = m = `-3/2`

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

y = `-(3x)/2 ± sqrt(4(9/4) - 5)`

= `-(3x)/2 ± 2`

∴ 2y = – 3x ± 4

∴ 3x + 2y = ± 4