Tangents to a Parabola

Strategies and Hints

  1. Line n has a slope of -1; line m, a slope of 1.

  2. To find the intersection of line m and the parabola, solve the system of equations y = x2 and y = x + b. This results in x2- x - b = 0.

  3. If a quadratic equation has just one solution, the discriminant must equal 0. Thus, the equation x2- x - b = 0 (which represents the single point of intersection of line m and the parabola) must have a discriminant of 0.

Solution

Solving x2- x - b = 0 gives a discriminant of 1 + 4b. Setting this equal 0 yields
b = . So, the coordinates of (a, b) are (0, ).