Question

If the chord joining the points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \) on the parabola \( y^2 = 12x \) subtends a right angle at the vertex of the parabola, then \( x_1x_2 - y_1y_2 \) is equal to

• A. 292
• B. 288
• C. 284
• D. 280

Hint:

For a parabola \( y^2 = 4ax \), if a chord subtends a right angle at the vertex, then the product of parameters satisfies \( t_1t_2 = -4 \).

Answer 1

The Correct Option is B

The given parabola is \[ y^2 = 12x \] which is of the standard form \( y^2 = 4ax \). Hence, \[ 4a = 12 \Rightarrow a = 3. \] The vertex of the parabola is at the origin \( O(0,0) \).
Step 1: Parametric coordinates of points on the parabola.
The parametric form of a point on the parabola \( y^2 = 4ax \) is \[ (at^2,\, 2at). \] Therefore, the coordinates of points \( P_1 \) and \( P_2 \) are \[ P_1(3t_1^2,\, 6t_1), \quad P_2(3t_2^2,\, 6t_2). \] Step 2: Condition for right angle at the vertex.
Since the chord \( P_1P_2 \) subtends a right angle at the vertex \( O \), we have \[ \overrightarrow{OP_1} \cdot \overrightarrow{OP_2} = 0. \] Thus, \[ (3t_1^2)(3t_2^2) + (6t_1)(6t_2) = 0. \] \[ 9t_1^2t_2^2 + 36t_1t_2 = 0. \] Dividing by 9, \[ t_1^2t_2^2 + 4t_1t_2 = 0. \] \[ t_1t_2(t_1t_2 + 4) = 0. \] Since the points are distinct, \[ t_1t_2 = -4. \] Step 3: Compute \( x_1x_2 - y_1y_2 \).
Using the parametric coordinates, \[ x_1x_2 = (3t_1^2)(3t_2^2) = 9t_1^2t_2^2, \] \[ y_1y_2 = (6t_1)(6t_2) = 36t_1t_2. \] So, \[ x_1x_2 - y_1y_2 = 9t_1^2t_2^2 - 36t_1t_2. \] Substituting \( t_1t_2 = -4 \), \[ x_1x_2 - y_1y_2 = 9(16) - 36(-4). \] \[ = 144 + 144 = 288. \] Final Answer: \[ \boxed{288} \]