Question

Let \(P(3\cos\alpha, 2\sin\alpha), \alpha \neq 0\), be a point on the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). \(Q\) be a point on the circle \(x^2 + y^2 - 14x - 14y + 82 = 0\) and \(R\) be a point on the line \(x + y = 5\) such that the centroid of the triangle \(PQR\) is \(\left( 2 + \cos\alpha, 3 + \frac{2}{3}\sin\alpha \right)\). Then the sum of the ordinates of all possible points \(R\) is:

• A. 6
• B. 2
• C. 4
• D. 8

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
By equating the given centroid coordinates to the centroid formula applied to vertices P, Q, and R, we can deduce a direct geometric locus for point Q. Finding the intersection of this locus with the given circle gives exact coordinates for Q, allowing us to find R.
Step 2: Key Formula or Approach:
Centroid \(G(x, y)\) of triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\):
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \] Step 3: Detailed Explanation:
Let \(Q = (x_1, y_1)\) and \(R = (x_2, y_2)\).
Equating the \(x\)-coordinates of the centroid:
\[ \frac{3\cos\alpha + x_1 + x_2}{3} = 2 + \cos\alpha \implies x_1 + x_2 = 6 \] Equating the \(y\)-coordinates:
\[ \frac{2\sin\alpha + y_1 + y_2}{3} = 3 + \frac{2}{3}\sin\alpha \implies y_1 + y_2 = 9 \] Since R lies on \(x + y = 5\), we have \(x_2 + y_2 = 5\).
Adding the derived coordinates of Q:
\[ x_1 + y_1 = (6 - x_2) + (9 - y_2) = 15 - (x_2 + y_2) = 15 - 5 = 10 \] So point Q lies on the line \(x + y = 10\).
Point Q also lies on the circle: \((x - 7)^2 + (y - 7)^2 = 16\) (completing the square).
Substitute \(y = 10 - x\) into the circle equation:
\[ (x - 7)^2 + (3 - x)^2 = 16 \] Let \(x - 5 = t\). Then \(x - 7 = t - 2\) and \(3 - x = -2 - t\).
\[ (t - 2)^2 + (-2 - t)^2 = 16 \implies 2t^2 + 8 = 16 \implies t^2 = 4 \implies t = \pm 2 \] If \(t = 2\), \(x_1 = 7\) and \(y_1 = 3\).
If \(t = -2\), \(x_1 = 3\) and \(y_1 = 7\).
We need the sum of the ordinates (\(y_2\)) of all possible points R.
Recall \(y_2 = 9 - y_1\).
For \(y_1 = 3 \implies y_2 = 6\).
For \(y_1 = 7 \implies y_2 = 2\).
The sum of all possible ordinates of R is \(6 + 2 = 8\).
Step 4: Final Answer:
The sum is 8.