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Question

If the circle C1:x2+y2=16 intersects another circle C2 of radius 5 in such a manner that common chord is of maximum length and has a slope equal to 34, then the absolute sum of coordinates of the centre C2 is

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Solution

We have C1:x2+y2=16
Centre O1(0,0), radius =4.
C2 is another circle with radius 5, let its centre O2 be (h,k).

Now, the common chord of circles C1 and C2 is of maximum length when chord is diameter of the smaller circle C1, and then it passes through centre O1 of circle C1.
Given that slope of this chord is 34.
Equation of AB is,
y=34x
3x4y=0 (1)

In right ΔAO1O2,
O1O2=5242=3
Also O1O2=r distance from (h,k) to (1)
3=∣ ∣3h4k32+42∣ ∣
±3=3h4k5
3h4k±15=0 (2)

Again, ABO1O2
mAB×mO1O2=1
34×kh=1
4h+3k=0 (3)
Solving 3h4k+15=0 and 4h+3k=0,
we get h=95, k=125
Again solving 3h4k15=0 and 4h+3k=0,
we get h=95, k=125
Thus the required centre is (95,125) or (95,125)
Absolute sum of coordinates of centre C2=35

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