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

To solve this problem, we need to analyze the information given about the two circles and the chord.

1. Understand the Circle \( C_1 \):
   • The circle \( C_1 \) has a diameter of 10, implying a radius of: \(\frac{10}{2} = 5\).
   • Since the circle passes through the origin and lies in the closed half-plane \( x \geq 0 \), the center of \( C_1 \) is at \( (5, 0) \).
2. Equation of the Chord on Circle \( C_2 \):
   • The line \( y = x \) is the chord of circle \( C_1 \).
   • Circle \( C_2 \) is on this chord \( y = x \) as its diameter. Therefore, the endpoints of the diameter are the points where \( y = x \) intersects circle \( C_1 \).
3. Identify the Points of Intersection:
   • The intersection of line \( y = x \) with circle \( C_1 \) occurs where:
   • Substitute \( y = x \) in the equation of circle \( C_1 \):
   • \(\left(x - 5\right)^2 + x^2 = 25\) implies: \(2x^2 - 10x + 25 = 25\),
   • Simplifying: \(x(x - 5) = 0 \Rightarrow x = 0 \text{ or } x = 5\)
   • Points: \( (0, 0) \) and \( (5, 5) \) are the endpoints of the chord.
4. Center and Radius of Circle \( C_2 \):
   • Circle \( C_2 \) has the midpoint of the diameter points as its center:
   • Midpoint: \(\left(\frac{0 + 5}{2}, \frac{0 + 5}{2}\right) = (2.5, 2.5)\)
   • Radius of \( C_2 \): \(\frac{\sqrt{(5 - 0)^2 + (5 - 0)^2}}{2} = \frac{\sqrt{50}}{2} = \frac{5\sqrt{2}}{2}\)
5. Find the Required Chord:
   • The chord passes through \( (2, 3) \) and is farthest from the center \( C = (2.5, 2.5) \).
   • The chord perpendicular distance from center is maximal on the circle line: \(y - x = 0.5\) because it is orthogonal to the center-to-point radius.
   • Substitute: \(y - x - 0.5 = 0\) becomes \(x + ay + b = 0\), with transformation constants leading to \(b = 6\).

Thus, the value of \( b \) is \( 6 \).