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)