Question

31. A circle C, of unit radius touches coordinate axes in first quadrant. Another circle C2 touch
C, externally and also touches each line of the pair xy – 4x – 4y +16 ​

Answer 1

Given:

A circle C, of unit radius touches coordinate axes in first quadrant.

Another circle C2 touch  C, externally and also touches each line of the pair xy – 4x – 4y +16 ​= 0

To find:

The radius of C2.

Solution:

Let the radius of circle C2 = r

From given, we have,

The radius of circle C1 = 1

xy – 4x – 4y + 16 ​= 0

x (y - 4) - 4(y - 4) = 0

(x - 4)(y - 4) = 0

x = 4 and y = 4 are the lines.

Now consider,

The diameters of circle C1, C2 and the lines x = 4 and y = 4 form a straight line.

Consider the attached while going through the following steps.

So, we get an equation,

OC1 + C1C2 + C2A = OA

√(1² + 1²) + (1 + r) + √(r² + r²) = √(4² + 4²)

√2 + (1 + r) + r√2 = 4√2

(1 + r) + r√2 = 3√2

r (1 + √2) = 3√2 - 1

r = (3√2 - 1) / (1 + √2)

rationalizing the above, we get,

r = 7 - 4√2

Hence the radius of circle C2 is 7 - 4√2