Intersections of Tangents and Normals

After finding a tangent or normal, use its line equation for any further geometry. A point where two objects meet must satisfy both equations; the derivative is no longer the equation you solve.

For $y=x^2$, the normal at $A=(1,1)$ is $y=-\tfrac12x+\tfrac32$. It meets the curve again at $B$.

The root $x=1$ is the known point $A$. “Meets again” asks for the other root, $x=-3/2$. Substituting into the curve gives $B=(-3/2,9/4)$; the normal gives the same y-value.

A tangent can also cross a curve at another point. Being a tangent describes its direction at the contact point, not how many intersections it has elsewhere.

• Two tangents or normals meet: solve their two line equations simultaneously.
• A line meets the x-axis: set $y=0$ in that line. For the y-axis, set $x=0$.
• A length or midpoint is requested: first find both complete endpoints, then use the coordinate formula.

For endpoints $(x_1,y_1),(x_2,y_2)$, the length is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. The midpoint averages each pair of coordinates. A triangle needs a base and its perpendicular height, not an arbitrary sloping side.

For a tangent with finite gradient, its angle to the positive x-axis satisfies $\tan\theta=m$: both ratios are vertical change divided by horizontal change. Use inverse tangent in degree mode, then choose the angle in the requested range.