Tangent of a Parabola

Question

Find the value of c such that y=x+c is a tangent to the parabola y=x^2-x-12.

Answer 1

The slope of the tangent line to the parabola at x is y' = 2x-1.

The slope of the line y = x + c is 1.

So, 2x - 1 = 1. Therefore, x = 1.

When x = 1, y = (1)² - 1 - 12 = -12

Since the line and parabola intersect when x = 1, we have 1 + c = -12.

So, c = -13