Question

The line \(x - y = 4\) is a chord of the circle \((x - 4)^2 + (y + 3)^2 = 9\), which cuts the circle at points Q & R. If \(P(\alpha, \beta)\) lies on the circle such that \(PQ = PR\), then find \((6\alpha + 8\beta)^2\).

Hint:

The condition $PQ=PR$ for points on a circle implies that $P$ lies on the diameter perpendicular to $QR$. Finding the equation of this diameter is much faster than solving for $Q$ and $R$ coordinates.

Answer 1

Correct Answer: 18

Step 1: Understanding the Concept:
In any circle, if a point \(P\) on the circumference is equidistant from two points \(Q\) and \(R\) on the circle (\(PQ = PR\)), then \(P\) must lie on the perpendicular bisector of the chord \(QR\). Since the perpendicular bisector of any chord passes through the center of the circle, the line \(CP\) (where \(C\) is the center) is perpendicular to the chord \(QR\).

Step 2: Key Formula or Approach:
1. Center of circle \(C(4, -3)\), Radius \(r = 3\). 2. Slope of chord \(QR\) (\(x - y = 4\)) is \(m_1 = 1\). 3. Slope of perpendicular line \(CP\) is \(m_2 = -1\).

Step 3: Detailed Explanation:
1. The equation of line \(CP\) passing through \((4, -3)\) with slope \(-1\): \[ y - (-3) = -1(x - 4) \implies y + 3 = -x + 4 \implies x + y = 1 \] 2. Since \(P(\alpha, \beta)\) lies on this line: \(\alpha + \beta = 1 \implies \beta = 1 - \alpha\). 3. \(P(\alpha, \beta)\) also lies on the circle: \[ (\alpha - 4)^2 + (1 - \alpha + 3)^2 = 9 \] \[ (\alpha - 4)^2 + (4 - \alpha)^2 = 9 \implies 2(\alpha - 4)^2 = 9 \] \[ (\alpha - 4)^2 = \frac{9}{2} \implies \alpha - 4 = \pm \frac{3}{\sqrt{2}} \] \[ \alpha = 4 \pm \frac{3}{\sqrt{2}}, \quad \beta = 1 - \alpha = -3 \mp \frac{3}{\sqrt{2}} \] 4. Calculate \(6\alpha + 8\beta\): \[ 6\alpha + 8\beta = 6\alpha + 8(1 - \alpha) = 8 - 2\alpha \] Substituting \(\alpha = 4 \pm \frac{3}{\sqrt{2}}\): \[ 8 - 2(4 \pm \frac{3}{\sqrt{2}}) = 8 - 8 \mp \frac{6}{\sqrt{2}} = \mp 3\sqrt{2} \] 5. Square the result: \[ (6\alpha + 8\beta)^2 = (\mp 3\sqrt{2})^2 = 9 \times 2 = 18 \]
Step 4: Final Answer:
The value of \((6\alpha + 8\beta)^2\) is 18.