(A) −6

(B) 10

(C) 6

(D) −2

Correct Answer: −2

Explanation

The circle C₁ has diameter 10 and passes through the origin. Since it lies in the half-plane x ≥ 0, the diameter lies along the x-axis.

End points of diameter of C₁ are (0, 0) and (10, 0)

Hence, the center and equation of circle C₁ are:

Center = (5, 0)

(x − 5)² + y² = 25

The chord y = x intersects this circle. Substitute y = x:

(x − 5)² + x² = 25

x² − 10x + 25 + x² = 25

2x² − 10x = 0

2x(x − 5) = 0

So the points of intersection are:

(0, 0) and (5, 5)

These two points form the diameter of the circle C₂.

Hence, center of C₂ is the midpoint:

C₂ center = ( (0+5)/2 , (0+5)/2 ) = (2.5, 2.5)

Now consider the point (2, 3). The chord of C₂ farthest from its center and passing through this point is perpendicular to the radius joining the center to this point.

Slope of the line joining center (2.5, 2.5) to (2, 3):

m = (3 − 2.5)/(2 − 2.5) = 0.5 / (−0.5) = −1

Therefore, slope of the required chord is the negative reciprocal:

m = 1

Equation of the line with slope 1 passing through (2, 3):

y − 3 = 1(x − 2)

y = x + 1

x − y + 1 = 0

Comparing with x + ay + b = 0:

a = −1,   b = 1

Hence,

a − b = −1 − 1 = −2

Therefore, the required value is −2.

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