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

k = - 8c.

Step-by-step 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.