Statement :1 If a parabola `y ^(2) = 4ax` intersects a circle in three co-normal points then the circle also passes through the vertr of the parabola. Because
Statement : 2 If the parabola intersects circle in four points `t _(1), t_(2), t_(3) and t_(4)` then `t _(1) + t_(2) + t_(3) +t_(4) =0` and for co-normal points `t _(1), t_(2) , t_(3)` we have `t_(1)+t_(2) +t_(3)=0.`

Options

A. Stateme-1 is True, Statement-2 is True, Statemetn-2 is correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is ture

Answer

To solve the problem, we need to analyze the statements regarding the intersection of a parabola and a circle, particularly focusing on the conditions of co-normal points.

### Step-by-Step Solution:

1. **Understanding the Parabola and Circle:**
The given parabola is \( y^2 = 4ax \). The vertex of this parabola is at the origin (0, 0). We need to consider a circle that intersects this parabola at three co-normal points.

2. **Definition of Co-normal Points:**
Co-normal points are points on a curve where the normals at those points intersect at a single point. For the parabola \( y^2 = 4ax \), if we denote the points of intersection with the circle as \( t_1, t_2, t_3, \) and \( t_4 \), the condition for co-normal points is given by:
\[ t_1 + t_2 + t_3 = 0 \]

3. **Using the Circle's Equation:**
The general equation of a circle can be written as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
We need to substitute the points of intersection from the parabola into this equation to find the conditions under which the circle passes through the vertex.

4. **Substituting Points:**
The points of intersection of the parabola with the circle can be expressed in terms of the parameter \( t \):
- For \( t_1 \): \( (at_1^2, 2at_1) \)
- For \( t_2 \): \( (at_2^2, 2at_2) \)
- For \( t_3 \): \( (at_3^2, 2at_3) \)
- For \( t_4 \): \( (at_4^2, 2at_4) \)

5. **Condition for the Circle to Pass Through the Vertex:**
Since the vertex of the parabola is at (0, 0), we need to check if substituting \( (0, 0) \) into the circle's equation yields a true statement:
\[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \implies c = 0 \]
This implies that the circle passes through the origin.

6. **Conclusion:**
Since we established that for the circle to intersect the parabola at three co-normal points, the condition \( t_1 + t_2 + t_3 = 0 \) must hold, and since the circle passes through the vertex of the parabola (0, 0), we can conclude that Statement 1 is true.

### Final Answer:
Both statements are correct. The circle passes through the vertex of the parabola when it intersects the parabola at three co-normal points.

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