Finding where a line meets a parabola, using the discriminant to tell secant, tangent and non-intersecting lines apart, and finding tangent lines.
Solving the system
To find where a line meets a parabola, solve the system. Substituting the line into the parabola gives a quadratic equation in \(x\). Its discriminant tells you how many intersection points there are.
One exception: a vertical line meets a parabola \(y = ax^2 + bx + c\) at exactly one point, but it isn't tangent.
Tangent lines
The condition "discriminant \(= 0\)" is the main tool for tangent lines.
Practice
Quick check
After substituting a line into a parabola, the quadratic has discriminant \(0\). The line is:
One solution means exactly one point of contact.
Quick check
How many points do the line \(x = 5\) and the parabola \(y = x^2 - 3\) have in common?
A vertical line meets the parabola at exactly one point, here \((5, 22)\), without being tangent.
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