Condition for the Line y = mx + c to be a Tangent to a Conic

(i) Parabola $y^2=4ax$. Let $y=mx+c$ be a tangent, touching at $(x_1,y_1)$. Comparing it with the point-form tangent $yy_1=2a(x+x_1)$ (same line, so coefficients proportional) gives $\dfrac{y_1}1=\dfrac{2a}m=\dfrac{2ax_1}c$, hence $y_1=\dfrac{2a}m$ and (substituting into $y_1^2=4ax_1$) $c=\dfrac am$. So

(ii) Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$. By the same coefficient-matching method, the condition is

(Only one of $y=mx+\sqrt{a^2m^2+b^2}$ or $y=mx-\sqrt{a^2m^2+b^2}$ is the tangent on a given side — not both simultaneously, for the same $m$.)

(iii) Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Likewise,

Results (proofs left to the reader, exactly as the textbook states them, but stated precisely here for use):

1. Two tangents can be drawn to a parabola, an ellipse, or a hyperbola from any external point (the slope condition, forced through the external point and squared, is always a quadratic in $m$).
2. Four normals can be drawn to an ellipse or a hyperbola from any external point (the normal-through-a-point condition is a quartic in the parameter). …