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Question
If y – 2x – k = 0 touches the conic 3x2 – 5y2 = 15, find the value of k.
Solution
Given line is y - 2x - k = 0
or y = 2x + k
Given conic section is 3x2 - 5y2 = 15
or `"x"^2/5 - "y"^2/3 = 1`
Here, m = 2, a2 = 5 and b2 = 3
Line y = mx + c touches the conic section
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` (Hyperbola)
`"y" = "mx"+- sqrt("a"^2"m"^2 - "b"^2)`
`"2x + "k" = 2"x" +- sqrt (5(2)^2 - 3)`
`"k" = +-sqrt(20-3)`
k = `+-sqrt17`
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