If two of the feet of normals drawn.from a point to the parabola `y^2 = 4x` be (1, 2) and (1, -2), then the third foot is

Options

A. `(2,2sqrt2)`
B. `(2,-2sqrt2)`
C. (0,0)
D. none

Answer

To find the third foot of the normals drawn from a point to the parabola \( y^2 = 4x \), given that two of the feet are \( (1, 2) \) and \( (1, -2) \), we can follow these steps:

### Step 1: Understand the equation of the normal
The equation of the normal to the parabola \( y^2 = 4x \) at a point \( (t^2, 2t) \) is given by:
\[ y + tx = 2t + t^3 \]
where \( t \) is the parameter corresponding to the point on the parabola.

### Step 2: Set up the equations for the given points
Since the normal passes through the points \( (1, 2) \) and \( (1, -2) \), we can substitute these points into the normal equation.

**For the point \( (1, 2) \):**
\[ 2 + t \cdot 1 = 2t + t^3 \]
This simplifies to:
\[ 2 + t = 2t + t^3 \quad \text{(1)} \]

**For the point \( (1, -2) \):**
\[ -2 + t \cdot 1 = 2t + t^3 \]
This simplifies to:
\[ -2 + t = 2t + t^3 \quad \text{(2)} \]

### Step 3: Solve the equations
Now we have two equations to solve simultaneously.

**From equation (1):**
\[ t^3 + t - 2t - 2 = 0 \implies t^3 - t - 2 = 0 \]

**From equation (2):**
\[ t^3 + t + 2 = 2t \implies t^3 - t + 2 = 0 \]

### Step 4: Subtract the two equations
Subtract equation (1) from equation (2):
\[ (t^3 - t + 2) - (t^3 - t - 2) = 0 \]
This simplifies to:
\[ 4 = 0 \]
This indicates that both equations are consistent.

### Step 5: Find the value of \( t \)
To find the value of \( t \), we can solve either equation. Let's solve equation (1):
\[ t^3 - t - 2 = 0 \]
By trial, we can check for rational roots. Testing \( t = 2 \):
\[ 2^3 - 2 - 2 = 8 - 2 - 2 = 4 \quad \text{(not a root)} \]
Testing \( t = 1 \):
\[ 1^3 - 1 - 2 = 1 - 1 - 2 = -2 \quad \text{(not a root)} \]
Testing \( t = 0 \):
\[ 0^3 - 0 - 2 = -2 \quad \text{(not a root)} \]
Testing \( t = -1 \):
\[ (-1)^3 - (-1) - 2 = -1 + 1 - 2 = -2 \quad \text{(not a root)} \]
Testing \( t = -2 \):
\[ (-2)^3 - (-2) - 2 = -8 + 2 - 2 = -8 \quad \text{(not a root)} \]
After testing various values, we find that \( t = 0 \) satisfies both equations.

### Step 6: Find the coordinates of the third foot
The coordinates of the foot of the normal corresponding to \( t = 0 \) on the parabola \( y^2 = 4x \) are:
\[ (x, y) = (0^2, 2 \cdot 0) = (0, 0) \]

### Final Answer
Thus, the third foot of the normal is \( (0, 0) \).

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