Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II

Equations of Tangent and Normal to Ellipse and Hyperbola

5.6.2

Equations of Tangent and Normal to Ellipse and Hyperbola

The derivations mirror §5.6.1's chord-limit method for the parabola (left to the reader, as the textbook notes); only the results are needed for problem-solving, so they are collected here for direct lookup.

Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ :

	Cartesian, at $(x_1,y_1)$	Parametric, at ' $\theta$ '
Tangent	$\dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}=1$	$\dfrac{x\cos\theta}a+\dfrac{y\sin\theta}b=1$
Normal	$\dfrac{a^2x}{x_1}-\dfrac{b^2y}{y_1}=a^2-b^2$	$\dfrac{ax}{\cos\theta}-\dfrac{by}{\sin\theta}=a^2-b^2$

Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ :

	Cartesian, at $(x_1,y_1)$	Parametric, at ' $\theta$ '
Tangent	$\dfrac{xx_1}{a^2}-\dfrac{yy_1}{b^2}=1$	$\dfrac{x\sec\theta}a-\dfrac{y\tan\theta}b=1$
Normal	$\dfrac{a^2x}{x_1}+\dfrac{b^2y}{y_1}=a^2+b^2$	$\dfrac{ax}{\sec\theta}+\dfrac{by}{\tan\theta}=a^2+b^2$ , i.e. $ax\cos\theta+by\cot\theta=a^2+b^2$

Start your 14-day free trial to unlock the full solution →

← Previous · 5.6.1Equation of Tangent and Normal to the Parabola y² = 4ax Next · 5.6.3 →Condition for the Line y = mx + c to be a Tangent to a Conic