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Question
Answer the following:
Find the equation of the tangent to the hyperbola x = 3 secθ, y = 5 tanθ at θ = `pi/3`
Solution
Given, equation of the hyperbola is
x = 3 sec θ, y = 5 tan θ.
Since sec2θ – tan2θ = 1,
`x^2/9 - y^2/25` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get
a2 = 9 and b2 = 25
∴ a = 3 and b = 5
Equation of tangent at P(θ) is
`(xsectheta)/"a" - (ytantheta)/"b"` = 1
∴ Equation of tangent at P`(pi/3)` is
`(xsec(pi/3))/3 - (ytan(pi/3))/5` = 1
∴ `(2x)/3 - (sqrt(3)y)/5` = 1
∴ `10x - 3sqrt(3)y` = 15
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