Question

If y – 2x – k = 0 touches the conic 3x² – 5y² = 15, find the value of k. 

Answer 1

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`