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  • The ellipse <img alt="\mathrm{\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7B%5Cmathrm%7Bx%7D%5E

The ellipse \mathrm{\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1} is rotated through a right angle in its own plane about its centre which is fixed. The locus of the point of intersection of the tangents to the ellipse in its original and in the new position is \mathrm{\left(\mathrm{x}^2+\mathrm{y}^2\right)\left(\mathrm{x}^2+\mathrm{y}^2-\mathrm{a}^2-\mathrm{b}^2\right)=\mathrm{k}\left(\mathrm{a}^2-\mathrm{b}^2\right) \mathrm{xy}}, where \mathrm{\mathrm{k}=}

Option: 1

\mathrm{K=1}


Option: 2

\mathrm{K=-1}


Option: 3

\mathrm{K=2}


Option: 4

\mathrm{K=-2}


Answers (1)

best_answer

The equation of the ellipse is \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}

Any point on it is \mathrm{P(a \cos \theta, b \sin \theta)}. The equation of the tangent is

\mathrm{ \frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1 }                                                             ......(1)
When the ellipse is rotated through a right angle, its equation becomes

\mathrm{ \frac{x^2}{b^2}+\frac{y^2}{a^2}=1 }

The point P also change its position to \mathrm{(-\mathrm{b} \sin \theta, \mathrm{a} \cos \theta)}

The equation of the tangent is \mathrm{-\frac{\mathrm{x}}{\mathrm{b}} \sin \theta+\frac{\mathrm{y}}{\mathrm{a}} \cos \theta=1}        ......(2)

Elimination of \mathrm{\theta} from (1) and (2) yields the locus of the point of intersection of the tangents (1) and (2). From (1) and (2), we find that
\mathrm{ \begin{aligned} & \cos \theta=\frac{(x+y) a}{x^2+y^2} \text { and } \sin \theta=\frac{(y-x) b}{x^2+y^2} \\\\ & \Rightarrow\left(x^2+y^2\right)^2=(x+y)^2 a^2+(y-x)^2 b^2=\left(x^2+y^2\right)\left(a^2+b^2\right)+2 x y\left(a^2-b^2\right) \end{aligned} }

Hence the locus of the point of intersection the two tangents is

\mathrm{\left(x^2+y^2\right)\left(x^2+y^2-a^2-b^2\right)=2\left(a^2-b^2\right) x y.}

Posted by

Irshad Anwar

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