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Question
Find k, if the line 3x + 4y + k = 0 touches 9x2 + 16y2 = 144
Solution
We know that y = mx + c will touch the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
if c2 = a2m2 + b2 ...(1)
The equation of the line is
3x + 4y + k = 0
∴ y = `-3/4 "x" -"k"/4`
Comparing this equation with y = mx + c, we get,
m = `-3/4`, c = `-"k"/4`
The equation of the ellipse is 9x2 + 16y2 = 144
∴ `x^2/16 + y^2/9` = 1
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
∴ a2 = 16, b2 = 9
Applying the tangency condition (1), we get,
`(-"k"/4)^2 = 16 xx (-3/4)^2 + 9`
∴ `"k"^2/16` = 9 + 9 = 18
∴ k2 = 16 × 9 × 2
∴ k = `± 12sqrt(2)`
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