Equation of Tangent to Ellipse in Reduced Form
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Theorem
Let $E$ be an ellipse embedded in a Cartesian plane in reduced form with the equation:
- $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_1, y_1}$ be a point on $E$.
The tangent to $E$ at $P$ is given by the equation:
- $\dfrac {x x_1} {a^2} + \dfrac {y y_1} {b^2} = 1$
Proof
From the slope-intercept form of a line, the equation of a line passing through $P$ is:
- $y - y_1 = \mu \paren {x - x_1}$
If this line passes through another point $\tuple {x_2, y_2}$ on $E$, the slope of the line is given by:
- $\mu = \dfrac {y_2 - y_1} {x_2 - x_1}$
Because $P$ and $Q$ both lie on $E$, we have:
\(\ds \dfrac {x_1^2} {a^2} + \dfrac {y_1^2} {b^2}\) | \(=\) | \(\ds 1 = \dfrac {x_2^2} {a^2} + \dfrac {y_2^2} {b^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {y_2^2} {b^2} - \dfrac {y_1^2} {b^2}\) | \(=\) | \(\ds \dfrac {x_1^2} {a^2} - \dfrac {x_2^2} {a^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\paren {y_2 + y_1} \paren {y_2 - y_1} } {b^2}\) | \(=\) | \(\ds \dfrac {\paren {x_1 + x_2} \paren {x_1 - x_2} } {a^2}\) | Difference of Two Squares | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {y_2 - y_1} {\paren {x_2 - x_1} }\) | \(=\) | \(\ds -\dfrac {b^2 \paren {x_1 + x_2} } {a^2 \paren {y_1 + y_2} }\) |
As $Q$ approaches $P$, we have that $y_2 \to y_1$ and $x_2 \to x_1$.
The limit of the slope is therefore:
- $-\dfrac {2 b^2 x_1} {2 a^2 y_1} = -\dfrac {b^2 x_1} {a^2 y_1}$
The equation of the tangent $\TT$ to $\CC$ passing through $\tuple {x_1, y_1}$ is therefore:
\(\ds y - y_1\) | \(=\) | \(\ds -\dfrac {b^2 x_1} {a^2 y_1}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {y_1} {b^2} \paren {y - y_1}\) | \(=\) | \(\ds -\dfrac {x_1} {a^2} \paren {x - x_1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {x x_1} {a^2} + \dfrac {y y_1} {b^2}\) | \(=\) | \(\ds \dfrac {x_1^2} {a^2} + \dfrac {y_1^2} {b^2}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | as $\tuple {x_1, y_1}$ is on $E$ |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $3$.