Question

Let the line \(x-y=4\) intersect the circle \(C:(x-4)^2+(y+3)^2=9\) at the points \(Q\) and \(R\). If \(P(\alpha,\beta)\) is a point on \(C\) such that \(PQ=PR\), then \((6\alpha+8\beta)^2\) is equal to ______.

Answer 1

Correct Answer: 144

Step 1: Understanding the Question
We are given a circle and a line that intersects it at two points, Q and R, forming a chord. We are looking for a point P on the circle that is equidistant from Q and R. Geometrically, this means the triangle PQR is isosceles. We need to find the coordinates of P and then evaluate the given expression.
Step 2: Key Formula or Approach
1. The locus of points equidistant from two points Q and R is the perpendicular bisector of the segment QR.
2. In a circle, the perpendicular bisector of any chord passes through the center of the circle.
3. Therefore, the point P must lie on a line that passes through the center of the circle and is perpendicular to the chord QR.
4. The product of the slopes of two perpendicular lines is -1, i.e., \( m_1 m_2 = -1 \).
Step 3: Detailed Explanation
The circle C is \( (x-4)^2 + (y+3)^2 = 9 \). Its center is \( O(4, -3) \) and its radius is \( r=3 \).
The line intersecting the circle is \( x - y = 4 \), which represents the chord QR. The slope of this line is \( m_{QR} = 1 \).
The condition \(PQ = PR\) means that the point P must lie on the perpendicular bisector of the chord QR.
The line containing P and the center O must be perpendicular to the chord QR.
The slope of the line OP must be \( m_{OP} = -1/m_{QR} = -1/1 = -1 \).
The line OP passes through the center \( O(4, -3) \) and has a slope of -1. Its equation is: \[ y - (-3) = -1(x - 4) \] \[ y + 3 = -x + 4 \] \[ x + y = 1 \] Since the point \( P(\alpha, \beta) \) lies on this line, its coordinates must satisfy the equation: \[ \alpha + \beta = 1 \implies \beta = 1 - \alpha \] The point \( P(\alpha, \beta) \) also lies on the circle C. So, its coordinates must satisfy the circle's equation: \[ (\alpha - 4)^2 + (\beta + 3)^2 = 9 \] Substitute \( \beta = 1 - \alpha \) into the circle equation: \[ (\alpha - 4)^2 + ((1 - \alpha) + 3)^2 = 9 \] \[ (\alpha - 4)^2 + (4 - \alpha)^2 = 9 \] \[ (\alpha - 4)^2 + (-( \alpha - 4))^2 = 9 \] \[ (\alpha - 4)^2 + (\alpha - 4)^2 = 9 \] \[ 2(\alpha - 4)^2 = 9 \implies (\alpha - 4)^2 = \frac{9}{2} \] We don't need to solve for \( \alpha \) and \( \beta \) individually. Let's try to evaluate the expression \( 6\alpha + 8\beta \).
Using \( \beta = 1 - \alpha \): \[ 6\alpha + 8\beta = 6\alpha + 8(1 - \alpha) = 6\alpha + 8 - 8\alpha = 8 - 2\alpha = 2(4 - \alpha) \] We need to find \( (6\alpha + 8\beta)^2 \), which is \( (2(4 - \alpha))^2 = 4(4 - \alpha)^2 \).
Since \( (\alpha - 4)^2 = (4 - \alpha)^2 \), we have: \[ 4(4 - \alpha)^2 = 4 \left( \frac{9}{2} \right) = 18 \] Step 4: Final Answer
The value of \( (6\alpha + 8\beta)^2 \) is 18.