Question

If the end points of chord of parabola \(y^2 = 12x\) are \((x_1, y_1)\) and \((x_2, y_2)\) and it subtend \(90^\circ\) at the vertex of parabola then \((x_1x_2 - y_1y_2)\) equals :

• A. 288
• B. 280
• C. 290
• D. not possible

Hint:

For a parabola \(y^2=4ax\), if a chord joining points with parameters \(t_1\) and \(t_2\) subtends a right angle at the vertex, the condition is always \(t_1t_2 = -4\). This is a very useful property to remember.

Answer 1

The Correct Option is A

Concept: 

The given equation of the parabola is \( y^2 = 12x \), and the end points of the chord are \( (x_1, y_1) \) and \( (x_2, y_2) \). The chord subtends an angle of \( 90^\circ \) at the vertex of the parabola.

Step 1: Equation of the Parabola

The equation of the parabola is: \[ y^2 = 12x \] which is a standard form for a parabola with its vertex at the origin \( (0, 0) \).

Step 2: Condition for Chord Subtending 90° at the Vertex

For a parabola, the condition for a chord subtending \( 90^\circ \) at the vertex is given by the formula: \[ (x_1 - x_2) \cdot (y_1 - y_2) = 288 \]

Step 3: Apply the Formula

Since the chord subtends \( 90^\circ \) at the vertex, the value of \( (x_1 - x_2)(y_1 - y_2) \) is \( 288 \).

Final Answer:

Thus, the value of \( (x_1 - x_2)(y_1 - y_2) \) equals: \[ \boxed{288} \]