Question

Let \( y = x \) be the equation of a chord of the circle \( C_1 \) (in the closed half-plane \( x \ge 0 \)) of diameter 10 passing through the origin. Let \( C_2 \) be another circle described on the given chord as diameter. If the equation of the chord of the circle \( C_2 \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_2 \), is \( x + ay + b = 0 \), then \( b \) is equal to:

• A. \( -2 \)
• B. \( 10 \)
• C. \( -6 \)
• D. \( 6 \)

Hint:

For maximum distance from the center, a chord passing through a fixed point must be perpendicular to the radius drawn to that point.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The circle \(C_1\) passes through the origin, lies in \(x \geq 0\), and has diameter 10.
We need to find its equation, its intersection with \(y=x\) to define circle \(C_2\), and then find a specific chord of \(C_2\).
Step 2: Key Formula or Approach:
The chord of a circle passing through a point \(P\) that is farthest from the center is the one perpendicular to the line joining the center to \(P\).
Step 3: Detailed Explanation:
Since the circle \(C_1\) has diameter 10, radius \(R = 5\).
Since it passes through \((0,0)\) and lies in \(x \geq 0\), its center must be \((5, 0)\).
Equation of \(C_1\): \((x - 5)^2 + y^2 = 25 \Rightarrow x^2 + y^2 - 10x = 0\).
Intersection with \(y = x\):
\(x^2 + x^2 - 10x = 0 \Rightarrow 2x^2 - 10x = 0 \Rightarrow x(x - 5) = 0\).
The points are \(A(0,0)\) and \(B(5,5)\).
\(AB\) is the diameter of circle \(C_2\).
Center of \(C_2\) is the midpoint of \(AB\): \(M = (\frac{5}{2}, \frac{5}{2})\).
Given point \(P(2, 3)\). The chord through \(P\) farthest from \(M\) is perpendicular to \(MP\).
Slope of \(MP = \frac{3 - 2.5}{2 - 2.5} = \frac{0.5}{-0.5} = -1\).
The slope of the required chord \(m = \frac{-1}{-1} = 1\).
Equation of the chord: \(y - 3 = 1(x - 2) \Rightarrow x - y + 1 = 0\).
Comparing with \(x + ay + b = 0\), we get \(a = -1\) and \(b = 1\).
Therefore, \(a - b = -1 - 1 = -2\).
Step 4: Final Answer:
The value of \(a - b\) is \(-2\).