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Question
Determine whether the line `x + 3ysqrt(2)` = 9 is a tangent to the ellipse `x^2/9 + y^2/4` = 1. If so, find the co-ordinates of the pt of contact
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Solution
Given equation of the ellipse is `x^2/9 + y^2/4` = 1.
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 9 and b2 = 4
Given equation of line is `x + 3ysqrt(2)` = 9,
i.e., y = `(-1)/(3sqrt(2)` x + `3/sqrt(2)`
Comparing this equation with y = mx + c, we get
m = `(-1)/(3sqrt(2)` and c = `3/sqrt(2)`
For the line y = mx + c to be a tangent to the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1, we must have
c2 = a2 m2 + b2
c2 = `(3/sqrt(2))^2 = 9/2`
a2 m2 + b2 = `9((-1)/(3sqrt(2)))^2 + 4`
= `1/2 + 4`
= `9/2`
= c2
∴ The given line is a tangent to the given ellipse and point of contact is
`((-"a"^2"m")/"c", "b"^2/"c") = (((-9)((-1)/(3sqrt(2))))/(3/sqrt(2)), 4/(3/sqrt(2)))`
= `(1, (4sqrt(2))/3)`.
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