Tangents To Ellipses

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As in circles and parabolas, the equation of a tangent to a given ellipse can take various different forms, all of which we discuss in this section. We will use the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) as our standard throughout this discussion.

TANGENTS AT P (x1, y1): Consider the ellipse

\[S(x,y):\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\]

and a point P (x1, y1) lying on this ellipse.

Thus,

\[\frac{x_{1}^{2}}{{{a}^{2}}}+\frac{y_{1}^{2}}{{{b}^{2}}}=1\qquad\qquad...\left( 1 \right)\]

The slope \({m_T}\)of the tangent at P (x1, y1) can be obtained by evaluating the derivative of the curve at P. For this purpose, we differentiate the equation of the ellipse:

\[\begin{align}&\frac{{2x}}{{{a^2}}} + \frac{{2y}}{{{b^2}}}\frac{{dy}}{{dx}} = 0\\& \Rightarrow \quad \frac{{dy}}{{dx}} = \frac{{ - {b^2}x}}{{{a^2}y}}\\& \Rightarrow \quad {m_T} = {\left. {\frac{{dy}}{{dx}}} \right|_{P({x_1},{y_1})}} = - \frac{{{b^2}{x_1}}}{{{a^2}{y_1}}}\end{align}\]

The equation of the tangent can now be obtained using point-slope form:

\[\begin{align}&y - {y_1} = \frac{{ - {b^2}{x_1}}}{{{a^2}{y_1}}}(x - {x_1})\\ &\Rightarrow\quad \frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = \frac{{x_1^2}}{{{a^2}}} + \frac{{y_1^2}}{{{b^2}}}\quad\quad...\left( 2 \right)\end{align}\]

Using (1), the RHS in (2) is 1 so that the equation of the ellipse is

\[\boxed{\frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = 1}\]

The equation obtained for the tangent can be, as in the case of circles and parabolas, written concisely in the form

\[\boxed{T({x_1},{y_1}) = 0}\]

TANGENT AT P\(\left( \theta  \right)\): If the point P is specified in parametric form instead of cartesian form, we simply substitute \({x_1} \to a\cos \theta ,\,\,{y_1} \to b\sin \theta \) in the equation of the tangent obtained above. Thus, the equation in this case is

\[\boxed{\frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = 1}\]

TANGENT OF SLOPE m: In example-15, we proved that any line of the form

\[\boxed{y = mx \pm \sqrt {{a^2}{m^2} + {b^2}} }\]

is a tangent to the ellipse \(\begin{align}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\end{align}\) whatever the value of m may be.

As an exercise, show that this tangent touches the ellipse at the point

\[\left( { \mp \frac{{{a^2}m}}{{\sqrt {{a^2}{m^2} + {b^2}} }},\,\, \pm \frac{{{b^2}}}{{\sqrt {{a^2}{m^2} + {b^2}} }}} \right)\]

Also show that from any point P, in general two tangents (real or imaginary) can be drawn to the ellipse (use the approach followed in Circles)

Download practice questions along with solutions for FREE:
Ellipses
grade 11 | Questions Set 1
Ellipses
grade 11 | Answers Set 1
Ellipses
grade 11 | Questions Set 2
Ellipses
grade 11 | Answers Set 2