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Question
Find the equations of the tangents to the hyperbola 5x2 – 4y2 = 20 which are parallel to the line 3x + 2y + 12 = 0
Solution
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
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