Question

Let \( O \) be the vertex of the parabola \( y^2 = 16x \). The locus of centroid of \( \triangle OPA \) when \( P \) lies on the parabola and \( A \) lies on the x-axis and \( \angle OPA = 90^\circ \).

• A. \( y^2 = 8(3x - 16) \)
• B. \( 9y^2 = 8(3x - 16) \)
• C. \( y^2 = 8(3x + 16) \)
• D. \( 9y^2 = 8(3x + 16) \)

Hint:

When the angle between two vectors is \( 90^\circ \), their dot product equals zero. This property can be used to find relationships between points in geometric problems.

Answer 1

The Correct Option is B

Step 1: Understanding the geometry of the problem.
We are given that the vertex of the parabola is \( O(0, 0) \) and the equation of the parabola is \( y^2 = 16x \). Let the coordinates of point \( P \) be \( (x_1, y_1) \) on the parabola, which satisfies the equation \( y_1^2 = 16x_1 \). Point \( A \) lies on the x-axis, so its coordinates are \( (x_2, 0) \). The centroid of \( \triangle OPA \) is given by: \[ G = \left( \frac{0 + x_1 + x_2}{3}, \frac{0 + y_1 + 0}{3} \right) = \left( \frac{x_1 + x_2}{3}, \frac{y_1}{3} \right) \]
Step 2: Conditions for \( \angle OPA = 90^\circ \).
For the angle \( \angle OPA = 90^\circ \), the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{PA} \) must be perpendicular. The direction of the vectors can be obtained from the coordinates of the points. The condition for two vectors to be perpendicular is: \[ \overrightarrow{OP} \cdot \overrightarrow{PA} = 0 \] Using the coordinates of \( O(0, 0) \), \( P(x_1, y_1) \), and \( A(x_2, 0) \), the equation becomes: \[ (x_1 - 0)(x_2 - x_1) + (y_1 - 0)(0 - y_1) = 0 \] \[ x_1 x_2 - x_1^2 - y_1^2 = 0 \] Substitute \( y_1^2 = 16x_1 \) from the equation of the parabola: \[ x_1 x_2 - x_1^2 - 16x_1 = 0 \]
Step 3: Simplify and find the relation.
We need to solve for the relationship between \( x_1 \) and \( x_2 \) (the coordinates of \( A \)) and eliminate \( x_1 \) in terms of \( x_2 \). After solving this equation, we obtain the equation for the locus of the centroid. The result is the equation \( 9y^2 = 8(3x - 16) \), which is option (2).