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  • If <img alt="\mathrm{ \frac{3}{2+e^{i \theta}}=\lambda x+i \mu y,}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Cfrac%7B3%7D%7B2&plus;e%5E%7Bi%20%5Ctheta%7D%7D%3D%5Cla

If \mathrm{ \frac{3}{2+e^{i \theta}}=\lambda x+i \mu y,} then locus of  \mathrm{P(x, y)} will represent  \mathrm{a / a n}

Option: 1

\mathrm{\text { Ellipse if } \lambda=1, \mu=2}


Option: 2

\mathrm{\text { Pair of straight lines if } \mu=0, \lambda=1}


Option: 3

\mathrm{\text { Circle if } \lambda=\mu=1}


Option: 4

\mathrm{\text { All of the above }}


Answers (1)

best_answer


\mathrm{ \begin{aligned} & \text 2+e^{i \theta}=\frac{3}{\lambda x+i \mu y} \\\\ & \Rightarrow 2+\cos \theta+i \sin \theta=\frac{3}{\lambda x+i \mu y} \\\\ & \Rightarrow \cos \theta+i \sin \theta=\frac{(3-2 \lambda x)-2 i \mu y}{\lambda x+i \cdot \mu y} \end{aligned} }
On taking modulus to both sides,

\mathrm{ \begin{aligned} & \Rightarrow 1=\frac{\sqrt{(3-2 \lambda x)^2+(-2 \mu y)^2}}{\sqrt{(\lambda x)^2+(\mu y)^2}} \\\\ & \Rightarrow \lambda^2 x^2+\mu^2 y^2=9-12 \lambda x+4 \lambda^2 x^2+4 \mu^2 y^2 \\\\ & \Rightarrow \lambda^2 x^2+\mu^2 y^2-4 \lambda x+3=0\, \, \, \, ....(A) \end{aligned} }

This is the locus of \mathrm{P(x, y)}
(a) If \mathrm{\lambda=1, \mu=2} then (A) becomes \mathrm{x^2+4 y^2-4 x+3=0}

\mathrm{ \Rightarrow(x-2)^2+4 y^2=1 \Rightarrow \frac{(x-2)^2}{1^2}+\frac{y^2}{1 / 4}=1 \text {, } }
which is an ellipse.

(b) If \mathrm{\mu=0, \lambda=1 \, \, then \, \, (A) \, \, becomes \, \, x^2-4 x+3=0}

\Rightarrow x=1,3; which represents a pair of straight lines.

(c) If  \mathrm{\lambda=\mu=1} then (A) becomes \mathrm{x^2+y^2-4 x+3=0}

\mathrm{\Rightarrow(x-2)^2+y^2=1}, which is a circle.

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mansi

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