If the tangents to the parabola `y^2=4ax`at the points `(x_1,y_1)`and (`(x_2,y_2)`meet at the point `(x_3,y_3)`then

Options

A. `y_3=sqrt(y_1y_2)`
B. `2y_3=y_1+y_2`
C. `2/y_3==1/y_1+1/y_2`
D. none of these

Answer

To solve the problem, we need to derive the relationship between the points where the tangents to the parabola \( y^2 = 4ax \) at points \( (x_1, y_1) \) and \( (x_2, y_2) \) meet at the point \( (x_3, y_3) \).

### Step-by-Step Solution:

1. **Identify the Points on the Parabola**:
The points \( (x_1, y_1) \) and \( (x_2, y_2) \) lie on the parabola \( y^2 = 4ax \). This means:
\[ y_1^2 = 4ax_1 \quad \text{and} \quad y_2^2 = 4ax_2 \]

2. **Equation of the Tangents**:
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point \( (x_1, y_1) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
Similarly, the equation of the tangent at the point \( (x_2, y_2) \) is:
\[ yy_2 = 2a(x + x_2) \]

3. **Finding the Intersection Point**:
To find the intersection point \( (x_3, y_3) \) of these two tangents, we can solve the two equations simultaneously.

From the first tangent:
\[ y = \frac{2a}{y_1}(x + x_1) \]
Substitute this expression for \( y \) into the second tangent equation:
\[ \left(\frac{2a}{y_1}(x + x_1)\right)y_2 = 2a(x + x_2) \]
Simplifying gives:
\[ \frac{2a y_2}{y_1}(x + x_1) = 2a(x + x_2) \]

4. **Eliminating \( a \)**:
Assuming \( a \neq 0 \), we can divide both sides by \( 2a \):
\[ \frac{y_2}{y_1}(x + x_1) = x + x_2 \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{y_2}{y_1}x + \frac{y_2}{y_1}x_1 = x + x_2 \]
\[ \left(\frac{y_2}{y_1} - 1\right)x = x_2 - \frac{y_2}{y_1}x_1 \]

6. **Finding \( x_3 \)**:
Solving for \( x \) gives:
\[ x_3 = \frac{x_2 - \frac{y_2}{y_1}x_1}{\frac{y_2}{y_1} - 1} \]

7. **Finding \( y_3 \)**:
Now, substituting \( x_3 \) back into either tangent equation to find \( y_3 \):
\[ y_3 = \frac{2a}{y_1}(x_3 + x_1) \]

8. **Final Relationship**:
The relationship between the points can be summarized as:
\[ x_3 = \frac{y_2}{y_1}x_1 + x_2 - \frac{y_2}{y_1}x_1 \]
This implies that the coordinates \( (x_3, y_3) \) can be expressed in terms of \( (x_1, y_1) \) and \( (x_2, y_2) \).