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Two points Pa n dQ are given. R is a ...

Two points `Pa n dQ` are given. `R` is a variable point on one side of the line `P Q` such that `/_R P Q-/_R Q P` is a positive constant `2alphadot` Find the locus of the point `Rdot`

Text Solution

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Let `R-=(x_1,y_1)`.
Also , let, `angle RPM =thet and angleRQM=phi`.

In `DeltaRMP,tantheta=(RM)/(MP)=(y_1)/(a-x_1)`
In `DeltaRQM, tantheta phi=(RM)/(QM)=(y_1)/(a+x_1)`
But given `angle RPQ-anlgeRQP=2alpha` (constant).Therefore, `theta -phi=2alpha`
or `tan(theta-phi)=tan2alpha`
or `(tantheta-tantheta)/(1+tanthetatantheta)=tan2alpha`
or `((y_(1))/(a-x_(1))-(y_(1))/(a+x_(1)))/(1+(y_(1))/(a-x_(1))(y_(1))/(a+x_(1)))=tan2alpha`
or `(2x_1y_1)/(a^2=x_1^2+y_1^2)=tanalpha`
or `a^2-x_1^(2)+y_1^(2)=2x_1y_1cot2alpha`
or `x_1^2-y_1^2+2x_1y_1cot2alpha=a^2`
Hence, the locus of the point `R(x_1,y_1)` is `x^2-y^2+2xycot2alpha=a^2`
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