If The Circle C1, x2 + y2 = 16 Intersects Another Circle C2 of Radius 5 In Such A Manner That The Common Chord Is Of Maximum Length And Has A Slope Equal To (3/4), Find The Co-ordinates Of Centre C2.

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Question

If the circle C1, x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to (3/4), find the co-ordinates of centre C2.

Solution

Correct option is

 

Let the centre of the circle C2 is Q(h, k), Equation of the circle C2 is

                    

or          x2 + y2 – 2xh – 2yk + h2 + k2 – 25 = 0  

              

And equation of circle   

∴ Equation of common chord is

                       C1 – C2 = 0  

  

or        2hx + 2ky – (h2 + k2 – 9) = 0                   …(1)

           Slope of this line = –h/k

But, it is given that its slope = 3/4

  

or                       3k + 4h = 0                        …(2)

Let P be the length perpendicular from P(0, 0) on chord (1), then

                              

Length of this chord AB = 2AM  

                                         

This chord has maximum length, then p = 0 then from (3), 

                             h2 + k2 – 9 = 0                        …(4)

Solving (2) and (4) we get 

                              

  

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