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