Question

If the 3x – 4y = k touches the hyperbola `x^2/5 - (4y^2)/5` = 1 then find the value of k

Answer 1

We know that y = mx + c will be a tangent to the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 if c² = a²m² – b² 

Given hyperbola is `x^2/5 - (4y^2)/5` = 1

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

∴ a² = 5, b² = `5/4`

Given tangent is 3x – 4y = k

∴ 4y = 3x –  k i.e. y = `3/4x - "k"/4`

∴ m = `3/4`, c = `-"k"/4`

Applying tangency condition c² = a²m² – b², we get,

`(-"k"/4)^2 = 5(3/4)^2 - 5/4`

∴ `"k"^2/16 = 5(9/16) - 5/4`

= `45/16 - 5/4`

= `25/16`

∴ k² = 25

∴ k = ±5.