The locus of centroid of triangle formed by a tangent to the parabola `y^(2) = 36x` with coordinate axes is

Options

A. `y^(2) =- 9x`
B. `y^(2) +3x = 0`
C. `y^(2) = 3x`
D. `y^(2) = 9x`

Explanation

To find the locus of the centroid of the triangle formed by a tangent to the parabola \( y^2 = 36x \) and the coordinate axes, we can follow these steps:

### Step 1: Equation of the Tangent
The standard form of the parabola is given by \( y^2 = 4ax \). Here, \( 4a = 36 \), so \( a = 9 \). The equation of the tangent to the parabola at point \( (t^2, 2at) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
Substituting \( a = 9 \) and \( x_1 = t^2 \) and \( y_1 = 2at = 18t \):
\[ yy_1 = 18(x + t^2) \]
This simplifies to:
\[ yy_1 = 18x + 18t^2 \]

### Step 2: Finding the Points of Intersection
To find the points where the tangent intersects the axes, we set \( y = 0 \) for the x-intercept:
\[ 0 = 18x + 18t^2 \implies x = -t^2 \]
For the y-intercept, set \( x = 0 \):
\[ yy_1 = 18(0 + t^2) \implies y = 18t \]
Thus, the points of intersection are \( (-t^2, 0) \) and \( (0, 18t) \).

### Step 3: Finding the Centroid
The centroid \( G \) of the triangle formed by the points \( (-t^2, 0) \), \( (0, 18t) \), and the origin \( (0, 0) \) is given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Substituting the coordinates:
\[ G\left( \frac{-t^2 + 0 + 0}{3}, \frac{0 + 18t + 0}{3} \right) = \left( -\frac{t^2}{3}, 6t \right) \]

### Step 4: Eliminating the Parameter
To find the locus, we need to eliminate the parameter \( t \). From the expression for \( G \):
1. Let \( x = -\frac{t^2}{3} \) which gives \( t^2 = -3x \).
2. Substitute \( t^2 \) into the expression for \( y \):
\[ y = 6t = 6\sqrt{-3x} \quad \text{(taking the positive root)} \]
Thus, squaring both sides:
\[ y^2 = 36(-3x) \implies y^2 = -108x \]

### Step 5: Final Locus Equation
The locus of the centroid is given by the equation:
\[ y^2 + 108x = 0 \]

### Conclusion
The locus of the centroid of the triangle formed by the tangent to the parabola \( y^2 = 36x \) with the coordinate axes is:
\[ y^2 + 108x = 0 \]