The number of distinct normals that can be drawn to the parabola `y^2 = 4x` from the point `(11/4,1/4)` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of distinct normals that can be drawn to the parabola \( y^2 = 4x \) from the point \( \left( \frac{11}{4}, \frac{1}{4} \right) \), we can follow these steps:

### Step 1: Understand the equation of the parabola
The given parabola is \( y^2 = 4x \). This is a standard form of a parabola that opens to the right.

### Step 2: Write 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:
\[ y - 2t = -\frac{1}{t}(x - t^2) \]
Rearranging this gives:
\[ y = -\frac{1}{t}x + \left(2t + \frac{t^2}{t}\right) = -\frac{1}{t}x + 3t \]

### Step 3: Substitute the point into the normal equation
We need the normal to pass through the point \( \left( \frac{11}{4}, \frac{1}{4} \right) \). Substituting \( x = \frac{11}{4} \) and \( y = \frac{1}{4} \) into the normal equation:
\[ \frac{1}{4} = -\frac{1}{t} \left( \frac{11}{4} \right) + 3t \]
Multiplying through by \( 4t \) to eliminate the fractions gives:
\[ t = -11 + 12t^2 \]
Rearranging this leads to:
\[ 12t^2 - 13t - 11 = 0 \]

### Step 4: Solve the quadratic equation
We can solve the quadratic equation \( 12t^2 - 13t - 11 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 12 \), \( b = -13 \), and \( c = -11 \):
\[ t = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 12 \cdot (-11)}}{2 \cdot 12} \]
Calculating the discriminant:
\[ t = \frac{13 \pm \sqrt{169 + 528}}{24} = \frac{13 \pm \sqrt{697}}{24} \]

### Step 5: Determine the number of distinct normals
The discriminant \( \sqrt{697} \) is positive, indicating that there are two distinct real solutions for \( t \). Each value of \( t \) corresponds to a distinct normal to the parabola from the given point.

### Conclusion
Thus, the number of distinct normals that can be drawn to the parabola \( y^2 = 4x \) from the point \( \left( \frac{11}{4}, \frac{1}{4} \right) \) is **2**.

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