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  • Which one of the following is the common tangent to the ellipses,  <img alt="\mathrm{\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1 \& \frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1 }" src="https://entrance

Which one of the following is the common tangent to the ellipses,  \mathrm{\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1 \& \frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1 } ?

Option: 1

\mathrm{a y=b x+\sqrt{a^4-a^2 b^2+b^4}}


Option: 2

\mathrm{b y=a x-\sqrt{a^4+a^2 b^2+b^4}}


Option: 3

\mathrm{a y=b x-\sqrt{a^4+a^2 b^2+b^4}}


Option: 4

\mathrm{b y=a x+\sqrt{a^4-a^2 b^2+b^5}}


Answers (1)

best_answer

Equation of a tangent to \mathrm{\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1}

\mathrm{y=m x \pm \sqrt{\left(a^2+b^2\right) m^2+b^2}}             \mathrm{....(1)}

If (1) is also a tangent to the ellipse  \mathrm{\frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1} then

\mathrm{\left(a^2+b^2\right) m^2+b^2=a^2 m^2+a^2+b^2}      \mathrm{\text { (using } c^2=a^2 m^2+b^2 \text { ) }}

\mathrm{b^2 m^2=a^2 \Rightarrow m^2=\frac{a^2}{b^2} \Rightarrow m= \pm \frac{a}{b}}

\mathrm{y= \pm \frac{a}{b} x \pm \sqrt{\left(a^2+b^2\right) \frac{a^2}{b^2}+b^2}}

\mathrm{b y= \pm a x \pm \sqrt{a^4+a^2 b^2+b^4}}

Although there can be four common tangents but only one of these appears in B

Posted by

Riya

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