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  • A circle cuts a chord of length 4a on the X-axis and passes through a point on the Y-axis, distant 2b from the origin. Then, the locus of the center of this circle, is <div class

A circle cuts a chord of length 4a on the X-axis and passes through a point on the Y-axis, distant 2b from the origin. Then, the locus of the center of this circle, is 

Option: 1

a parabola


Option: 2

an ellipse


Option: 3

a straight line

 


Option: 4

a hyperbola


Answers (1)

best_answer

According to given information, we have the following figure

Let the equation of circle be

x^2+y^2+2 g x+2 f y+c=0 \, ............(i)

According the problem,

  4 a=2 \sqrt{g^2-c}  —-----------------(ii)

[ \because  The length of intercepts made by the circle  x^{2}+y^{2}+2gx+2fy+c=0  with 

X-axis is  2 \sqrt{g^2-c}    

Also, as the circle is passing through

\begin{aligned} & \therefore o+4 b^2+0+4 b f+c=0 \\ \Rightarrow & 4 b^2+4 b f+c=0 \end{aligned}

So, locus of is 

\begin{aligned} & 4 b^2-4 b f+x^2-4 a^2=0 \\ \Rightarrow & x^2=4 b y+4 a^2-4 b^2 \end{aligned}

which is a parabola.

 

Posted by

himanshu.meshram

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