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  • The tangents drawn at those points of the ellipse <img alt="\mathrm{\frac{x^2}{a}+\frac{y^2}{b}=(a+b)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%7D%7Ba%7D&am

The tangents drawn at those points of the ellipse \mathrm{\frac{x^2}{a}+\frac{y^2}{b}=(a+b)}, where it is cut by any tangent to  \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1} intersect at.

Option: 1

75^{\circ}


Option: 2

90^{\circ}


Option: 3

60^{\circ}


Option: 4

120^{\circ}


Answers (1)

best_answer

The given ellipses are
\mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 }......................(1)
\mathrm{\frac{x^2}{a(a+b)}+\frac{y^2}{b(a+b)}=1 }.....................(2)

Chord of contact of \mathrm{\left(x_1, y_1\right) } w.r.t. (2) ellipse is
\mathrm{\begin{aligned} & \frac{x x_1}{a(a+b)}+\frac{y y_1}{b(a+b)}=1 \\ & \text { or } \quad \mathrm{lx}+\mathrm{my}=\mathrm{n} \text { (say) } \\ & \text { If it touches } \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ & \end{aligned} }
then \mathrm{a^2 1^2+b^2 m^2=n^2 \quad from corollary\} }
\mathrm{\begin{array}{ll} \Rightarrow \quad & \frac{a^2 x_1^2}{a^2(a+b)^2}+\frac{b^2 y_1^2}{b^2(a+b)^2}=1 \\ \Rightarrow \quad & x_1^2+y_1^2=(a+b)^2 \end{array}}
locus of T \mathrm{\left(x_1, y_1\right)} is
\mathrm{x^2+y^2=(a+b)^2}
 

Posted by

Pankaj Sanodiya

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