The tangent to the parabola `y^(2)=4ax` at `P(at_(1)^(2), 2at_(1))" and Q"(at_(2)^(2), 2at_(2))` intersect on its axis, them

Options

A. `t_(1)=t_(2)`
B. `t_(1)=-t_(2)`
C. `t_(1)t_(2)=2`
D. `t_(1)t_(2)=-1`

Explanation

To solve the problem, we need to find the condition under which the tangents to the parabola \( y^2 = 4ax \) at points \( P(at_1^2, 2at_1) \) and \( Q(at_2^2, 2at_2) \) intersect on the axis of the parabola.

### Step-by-Step Solution:

1. **Identify the Parabola and Points**:
The given parabola is \( y^2 = 4ax \). The points \( P \) and \( Q \) on the parabola are given as \( P(at_1^2, 2at_1) \) and \( Q(at_2^2, 2at_2) \).

2. **Find the Equation of the Tangent at Point P**:
The equation of the tangent to the parabola \( y^2 = 4ax \) at the point \( (at_1^2, 2at_1) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
Substituting \( y_1 = 2at_1 \) and \( x_1 = at_1^2 \):
\[ y(2at_1) = 2a(x + at_1^2) \]
Simplifying this gives:
\[ 2aty = 2ax + 2a^2t_1^2 \]
Dividing by \( 2a \) (assuming \( a \neq 0 \)):
\[ ty = x + at_1^2 \]
Rearranging gives:
\[ x - ty + at_1^2 = 0 \quad \text{(Equation 1)} \]

3. **Find the Equation of the Tangent at Point Q**:
Similarly, for point \( Q(at_2^2, 2at_2) \), the tangent is:
\[ yy_2 = 2a(x + x_2) \]
Substituting \( y_2 = 2at_2 \) and \( x_2 = at_2^2 \):
\[ 2aty = 2ax + 2a^2t_2^2 \]
Dividing by \( 2a \):
\[ ty = x + at_2^2 \]
Rearranging gives:
\[ x - ty + at_2^2 = 0 \quad \text{(Equation 2)} \]

4. **Find the Intersection of the Two Tangents**:
To find the intersection of the two tangents, we can set the equations equal to each other:
\[ x - ty + at_1^2 = 0 \]
\[ x - ty + at_2^2 = 0 \]
Subtracting these two equations gives:
\[ at_1^2 - at_2^2 = 0 \]
This implies:
\[ at_1^2 = at_2^2 \]

5. **Condition for Intersection on the Axis**:
Since the tangents intersect on the axis of the parabola, which is the line \( x = 0 \), we substitute \( x = 0 \) into either equation:
\[ 0 - ty + at_1^2 = 0 \implies ty = at_1^2 \]
For the intersection to occur at the axis, the y-coordinate must also be zero. Therefore:
\[ t(y) = 0 \implies t_1 + t_2 = 0 \]
This leads to the conclusion:
\[ t_2 = -t_1 \]

### Final Condition:
Thus, the required condition is:
\[ t_2 = -t_1 \]