Name: Conic Sections and Ellipse
Uploaded: 2019-07-19
Duration: 3 min 55 s
Description: In this video, we find the equation of an ellipse that is centred at the origin when the eccentricity and vertices are given. Study this video for a better understanding.

Question

If the normals to the ellipse at points

(x_{1}, y_{1}), (x_{2}, y_{2})

and

(x_{3}, y_{3})

be concurrent, show that

|\begin{array}{l} x_{1} & y_{1} & x_{1} y_{1} \\ x_{2} & y_{2} & x_{2} y_{2} \\ x_{3} & y_{3} & x_{3} y_{3} \end{array}| = 0

.

Accepted Answer

Normal to ellipse at point $(x_{1}, y_{1})$ is given by $\frac{a^{2} x}{x_{1}} - \frac{b^{2} y}{y_{1}} = a^{2} - b^{2}$

Similarly, normal to ellipse at point $(x_{2}, y_{2})$ is given by $\frac{a^{2} x}{x_{2}} - \frac{b^{2} y}{y_{2}} = a^{2} - b^{2}$ and normal to ellipse at point $(x_{3}, y_{3})$ is $\frac{a^{2} x}{x_{3}} - \frac{b^{2} y}{y_{3}} = a^{2} - b^{2}$ .

Now, we know that three lines $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ are concurrent, if $|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = 0$ .

$\Rightarrow$ The normals to the ellipse at the three point will be concurrent if, $|\begin{matrix} \frac{a^{2}}{x_{1}} & \frac{- b^{2}}{y_{1}} & b^{2} - a^{2} \\ \frac{a^{2}}{x_{2}} & \frac{- b^{2}}{y_{2}} & b^{2} - a^{2} \\ \frac{a^{2}}{x_{3}} & \frac{- b^{2}}{y_{3}} & b^{2} - a^{2} \end{matrix}| = 0$

$\Rightarrow - a^{2} b^{2} (b^{2} - a^{2}) |\begin{matrix} \frac{1}{x_{1}} & \frac{1}{y_{1}} & 1 \\ \frac{1}{x_{2}} & \frac{1}{y_{2}} & 1 \\ \frac{1}{x_{3}} & \frac{1}{y_{3}} & 1 \end{matrix}| = 0$

$\Rightarrow |\begin{matrix} \frac{1}{x_{1}} & \frac{1}{y_{1}} & 1 \\ \frac{1}{x_{2}} & \frac{1}{y_{2}} & 1 \\ \frac{1}{x_{3}} & \frac{1}{y_{3}} & 1 \end{matrix}| = 0 (∵ - a^{2} b^{2} (b^{2} - a^{2}) \neq 0)$

Now, applying $R_{1} \to (x_{1} y_{1}) R_{1}, R_{2} \to (x_{2} y_{2}) R_{2}, R_{3} \to (x_{3} y_{3}) R_{3}$ we get,

$\Rightarrow |\begin{matrix} y_{1} & x_{1} & x_{1} y_{1} \\ y_{1} & x_{2} & x_{2} y_{2} \\ y_{3} & x_{3} & x_{3} y_{3} \end{matrix}| = 0$

Now, applying $C_{1} \leftrightarrow C_{2}$ , we get $|\begin{matrix} x_{1} & y_{1} & x_{1} y_{1} \\ x_{2} & y_{2} & x_{2} y_{2} \\ x_{3} & y_{3} & x_{3} y_{3} \end{matrix}| = 0$

Hence the result

If the normals to the ellipse at points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ be concurrent, show that $|\begin{array}{l} x_{1} & y_{1} & x_{1} y_{1} \\ x_{2} & y_{2} & x_{2} y_{2} \\ x_{3} & y_{3} & x_{3} y_{3} \end{array}| = 0$ .

Important Questions on Conic Section

Learn and Prepare for any exam you want !

If the normals to the ellipse at points x1,y1,x2,y2 and x3,y3 be concurrent, show that x1y1x1y1x2y2x2y2x3y3x3y3=0.

Important Questions on Conic Section

Learn and Prepare for any exam you want !

If the normals to the ellipse at points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ be concurrent, show that $|\begin{array}{l} x_{1} & y_{1} & x_{1} y_{1} \\ x_{2} & y_{2} & x_{2} y_{2} \\ x_{3} & y_{3} & x_{3} y_{3} \end{array}| = 0$ .