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\newcommand{\ds}{\displaystyle}
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{\bf Question}

Find a condition on $l, m, n$ to ensure that the line $lx + my + n
= 0$ is tangent to the ellipse $\ds \frac{x^2}{a^2} +
\frac{y^2}{b^2} = 1.$

\vspace{.25in}

{\bf Answer}

The equation of the tangent at $(x_0, y_0)$ is $$\frac{xx_0}{a^2}
+ \frac{y y_0}{b^2}=1$$  This is the same as the line
$\ds-\frac{l}{n} x - \frac{m}{n}y = 1$ if and only if $$
\frac{x_0}{a^2} = -\frac{l}{n} \hspace{.2in} \frac{y_0}{b^32} =
-\frac{m}{n}$$ $${\rm i.e.\ \ }x_0 = -\frac{l}{a^2}{n}
\hspace{.2in} y_0 = -\frac{mb^2}{n}$$

But $(x_0,y_0)$ lies on the ellipise, so $$1 = \frac{x_0^2}{a^2} +
\frac{y_0^2}{b^2} = \frac{l^2a^4}{n^2a^2} + \frac{m^2b^2}{n^2b^2}
 {\rm \ \ i.e.\ \ } l^2a^2 + m^2b^2 = n^2$$  




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