Consider a point P on a parabola such that 2 of the normal drawn from it to the parabola are at right angles on parabola, then
If `P -= (x _(1), y _(1)),` the slope of third normal is, if If the equation of parabola is `y^(2)= 8x`

Options

A. `(y _(1))/(8)`
B. `(y _(1))/(2)`
C. `-(y _(1))/(8)`
D. `- (y _(1))/(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the equation of the parabola
The given equation of the parabola is \( y^2 = 8x \). This can be rewritten in the standard form \( y^2 = 4ax \), where \( 4a = 8 \). Thus, we find:
\[ a = 2 \]

**Hint:** Recall that the standard form of a parabola is \( y^2 = 4ax \).

### Step 2: Write the 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:
\[ y = mx - 2am - am^3 \]
Substituting \( a = 2 \) into the equation gives:
\[ y = mx - 4m - 2m^3 \]

**Hint:** Remember that the slope \( m \) is the slope of the normal line.

### Step 3: Substitute the point \( (x_1, y_1) \)
Since the normal passes through the point \( (x_1, y_1) \), we can substitute these coordinates into the normal equation:
\[ y_1 = mx_1 - 4m - 2m^3 \]
Rearranging this gives:
\[ 2m^3 + 4 - x_1m + y_1 = 0 \]

**Hint:** This equation is a cubic in terms of \( m \).

### Step 4: Analyze the cubic equation
This cubic equation has three roots \( m_1, m_2, m_3 \) corresponding to the slopes of the normals. We know that two of these normals are at right angles, which means:
\[ m_1 m_2 = -1 \]

**Hint:** The product of the slopes of two perpendicular lines is \(-1\).

### Step 5: Use the relationship of the roots
The product of the roots of a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by:
\[ m_1 m_2 m_3 = -\frac{d}{a} \]
In our case, we have:
\[ m_1 m_2 m_3 = -\frac{y_1}{2} \]
Substituting \( m_1 m_2 = -1 \) into this gives:
\[ (-1) m_3 = -\frac{y_1}{2} \]
Thus, we can solve for \( m_3 \):
\[ m_3 = \frac{y_1}{2} \]

**Hint:** Use the relationships between the roots of the polynomial to find the unknown slope.

### Final Answer
The slope of the third normal is:
\[ \boxed{\frac{y_1}{2}} \]