Question

Given that the line y=c-2x is tangent to the curve y^2=kx where c and k are non-zero constants,express k in terms of c

Answer 1

k = -8c.

Explanation:

y^2 = kx

Find the slope of the tangent by differentiating:

y' 2y = k

y' = k / 2y = the slope of the tangent.

The given equation of the tangent is y = -2x + c so the slope = -2.

Therefore k/2y = -2 so k = -4y and y = -k/4

y^2 = kx so k^2/16 = kx giving x = k/16.

Substituting for x and y in y = -2x + c:

-k/4 = -2 k/16 + c

-k/4 = -k/8 + c

c = -k/4 + k/8 = -k/8.

So -k = 8c

k = -8c.