Mathematico | Mathematico

If a line and a parabola intersect exactly once, the line is said to be a tangent to the parabola.

Given that the line with equation

y = x + k

is a tangent to the parabola with equation

y = x^{2} - 2 x + 2

find the value of $504 k$

Hint

The equation

x^{2} - 2 x + 2 = x + k

has exactly one solution.

Hint

So the discriminant of

x^{2} - 2 x + 2 = x + k

must be $0$ .

Solution

The equation

x^{2} - 2 x + 2 = x + k

has exactly one solution, so the discriminant of

\begin{aligned} x^{2} - 2 x + 2 & = x + k \\ x^{2} - 3 x + (2 - k) & = 0 \end{aligned}

must be $0$ . Thus

\begin{aligned} 9 - 4 (2 - k) & = 0 \\ 1 + 4 k & = 0 \\ k & = - \frac{1}{4} \end{aligned}