Circle 3 Question 22

22. Two circles, each of radius 5 units, touch each other at (1,2). If the equation of their common tangent is 4x+3y=10, find the equations of the circles. (1991,4M)

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Answer:

Correct Answer: 22. (x52) + (y52) = 52 and (x+32) + (y+12) = 52

Solution:

  1. We have,

Slope of the common tangent =43

 Slope of C1C2=34

If C1C2 makes an angle θ with X-axis, then cosθ=45 and sinθ=35.

So, the equation of C1C2 in parametric form is

x14/5=y23/5

Since, C1 and C2 are points on Eq. (i) at a distance of 5 units from P.

So, the coordinates of C1 and C2 are given by

x14/5=y23/5=±5x=1±4

 and y=2±3

Thus, the coordinates of C1 and C2 are (5,5) and (3,1), respectively.

Hence, the equations of the two circles are

and

(x5)2+(y5)2=52(x+3)2+(y+1)2=52



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