Let the latus rectum of the parabola y 2 = 4 x be the common chord to the circles C 1 and C 2 each of them having radius 2 5 . Then, the distance between the centres of the circles C 1 and C 2 is :

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Aicharya · Accepted Answer

Co-ordinates of focus of parabola will be $S = (a, 0) = (1, 0)$ .

Now, length of latus-rectum $= 4$ .

So, length of semi latus-rectum $= 2$ .

Also, $∠ L S C_{1} = 90 °$

Now, radius of circle $L C_{1} = 2 \sqrt{5}$

Now, applying pythagoras theorem in $△ L S C_{1}$ ,

${(L C_{1})}^{2} = {(C_{1} S)}^{2} + {(L S)}^{2}$

$\Rightarrow {(C_{1} S)}^{2} = {(L C_{1})}^{2} - {(L S)}^{2} = {(2 \sqrt{5})}^{2} - {(2)}^{2} = 16$

$\Rightarrow C_{1} S = \sqrt{16} = 4$

$∴ C_{1} C_{2} = 2 C_{1} S = 2 \times 4 = 8$ .

Let the latus rectum of the parabola y 2 = 4 x be the common chord to the circles C 1 and C 2 each of them…

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