The set of points on the axis of the parabola `y^(2)-2y-4x+5=0` from which all the three normals drawn to the parabola are real and distinct, is

Options

A. `{(x,1): x gt 3}`
B. `{(x, -1): x ge1}`
C. `{(x, 3):x ge 1}`
D. `{(x, -3): x ge 3}`

Answer

To solve the problem of finding the set of points on the axis of the parabola \(y^2 - 2y - 4x + 5 = 0\) from which all three normals drawn to the parabola are real and distinct, we will follow these steps:

### Step 1: Rewrite the parabola in standard form
We start with the given equation of the parabola:
\[ y^2 - 2y - 4x + 5 = 0 \]
Rearranging gives:
\[ y^2 - 2y = 4x - 5 \]
Next, we complete the square for the left-hand side:
\[ (y^2 - 2y + 1) = 4x - 5 + 1 \]
This simplifies to:
\[ (y - 1)^2 = 4x - 4 \]
or
\[ (y - 1)^2 = 4(x - 1) \]
This is now in the standard form of a parabola, \( (y - k)^2 = 4a(x - h) \), where the vertex is at \((1, 1)\) and the axis of the parabola is \(y = 1\).

### Step 2: Find the equation of the normal
The equation of the normal to the parabola at a point \((x_0, y_0)\) can be expressed as:
\[ y - y_0 = -\frac{1}{m}(x - x_0) \]
where \(m\) is the slope of the tangent at that point.

For the parabola \( (y - 1)^2 = 4(x - 1) \), the slope of the tangent at any point can be derived from the derivative. The normal can be expressed in terms of the slope \(m\) as:
\[ y - 1 = m(x - 1) - 2m(m^2) \]
This gives us the relationship between \(x\) and \(m\).

### Step 3: Determine conditions for real and distinct normals
Assuming a point on the axis of the parabola as \((h, 1)\), we substitute \(y = 1\) into the normal equation:
\[ 1 - 1 = m(h - 1) - 2m(m^2) \]
This simplifies to:
\[ 0 = m(h - 1) - 2m^3 \]
Factoring out \(m\):
\[ m[(h - 1) - 2m^2] = 0 \]
This implies either \(m = 0\) or \((h - 1) - 2m^2 = 0\).

### Step 4: Solve for \(h\)
From \((h - 1) - 2m^2 = 0\), we can express \(h\) in terms of \(m\):
\[ h - 1 = 2m^2 \implies h = 2m^2 + 1 \]
For \(m\) to yield three distinct normals, the expression \(2m^2 + 1\) must be greater than 3 (since \(h\) must be greater than 3):
\[ 2m^2 + 1 > 3 \implies 2m^2 > 2 \implies m^2 > 1 \implies m > 1 \text{ or } m < -1 \]

### Step 5: Conclusion
Thus, for all three normals to be real and distinct, the condition we derived is:
\[ h > 3 \]
Therefore, the set of points on the axis of the parabola from which all three normals drawn to the parabola are real and distinct is:
\[ \boxed{(h, 1) \text{ where } h > 3} \]