Consider a point P on a parabola such that 2 of the normal drawn from it to the parabola are at night angles on parabola, then
The ratio of latus rectum of given parabola and that of made by locus of point P is

Options

A. `4:1`
B. `2:1`
C. `16:1`
D. `1:1`

Answer

To solve the problem, we need to find the ratio of the latus rectum of a given parabola and that of the parabola formed by the locus of a point \( P \) from which two normals are drawn at right angles to the original parabola.

### Step-by-step Solution:

1. **Identify the Given Parabola**:
The given parabola is \( y^2 = 4ax \).

2. **Calculate the Latus Rectum of the Given Parabola**:
The latus rectum \( L \) of the parabola \( y^2 = 4ax \) is given by the formula:
\[ L = 4a \]
For this problem, we can assume \( a = 1 \) (as per the video transcript), thus:
\[ L = 4 \times 1 = 4 \]

3. **Equation of the Normal to the Parabola**:
The equation of the normal to the parabola at a point \( (x_1, y_1) \) is given by:
\[ y - y_1 = -\frac{y_1}{2a}(x - x_1) \]
For our parabola, this can be rewritten as:
\[ mx - 2a m - a m^3 = 0 \]
where \( m \) is the slope of the normal.

4. **Assume Point \( P \)**:
Let \( P(h, k) \) be a point from which two normals are drawn to the parabola. The normals are at right angles, which means the slopes of the normals \( m_1 \) and \( m_2 \) satisfy:
\[ m_1 m_2 = -1 \]

5. **Form a Cubic Equation**:
The normal equations yield a cubic equation in terms of \( m \):
\[ m^3 + (2a - h)m + k = 0 \]
The roots of this equation are \( m_1, m_2, m_3 \).

6. **Using the Product of Roots**:
From Vieta's formulas, the product of the roots \( m_1 m_2 m_3 \) is given by:
\[ m_1 m_2 m_3 = -\frac{k}{a} \]
Since \( m_1 m_2 = -1 \), we have:
\[ -1 \cdot m_3 = -\frac{k}{a} \implies m_3 = \frac{k}{a} \]

7. **Substituting Back into the Normal Equation**:
Substitute \( m_3 \) back into the cubic equation:
\[ k^2 + (2a - h)k + a^2 = 0 \]

8. **Finding the Locus**:
The locus of point \( P \) can be derived from the equation:
\[ k^2 = a(h - 3a) \]
This gives us the equation of the locus:
\[ y^2 = a(x - 3a) \]

9. **Identify the New Parabola**:
The new parabola can be written as:
\[ y^2 = 4a' x \]
where \( a' = \frac{a}{4} \).

10. **Calculate the Latus Rectum of the New Parabola**:
The latus rectum of the new parabola is:
\[ L' = 4a' = 4 \left(\frac{a}{4}\right) = a \]

11. **Finding the Ratio of Latus Rectum**:
Now we can find the ratio of the latus rectum of the original parabola to that of the new parabola:
\[ \text{Ratio} = \frac{L}{L'} = \frac{4a}{a} = 4 \]

Thus, the final answer is:
\[ \text{The ratio of the latus rectum of the given parabola and that of the locus of point P is } 4:1. \]