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

Options

A. `lamda,0),xgt1`
B. `(lamda,1),lamdagt3`
C. `(lamda,2),lamdagt6`
D. `(lamda,3),lamdagt8`

Answer

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

### Step 1: Rewrite the equation of the parabola
We start with the given equation of the parabola:
\[ y^2 - 4x - 2y + 5 = 0 \]
We can rearrange this equation to complete the square for the \(y\) terms:
\[ y^2 - 2y + 5 - 4x = 0 \]
Completing the square for \(y\):
\[ (y - 1)^2 - 1 + 5 - 4x = 0 \]
This simplifies to:
\[ (y - 1)^2 = 4x - 4 \]
or
\[ (y - 1)^2 = 4(x - 1) \]
This shows that the parabola opens to the right with vertex at \((1, 1)\).

### Step 2: Change of coordinates
Let’s shift the origin to the vertex \((1, 1)\). We define new coordinates:
\[ X = x - 1, \quad Y = y - 1 \]
Thus, the equation of the parabola in the new coordinates becomes:
\[ Y^2 = 4X \]

### Step 3: Equation of the normal to the parabola
The equation of the normal to the parabola \(Y^2 = 4X\) at a point where the slope of the tangent is \(m\) is given by:
\[ Y = mX - 2m + m^3 \]
We need to find the points on the x-axis (where \(Y = 0\)) from which all three normals are real.

### Step 4: Set the normal line to pass through the x-axis
Setting \(Y = 0\) in the normal equation gives:
\[ 0 = mX - 2m + m^3 \]
Rearranging this, we have:
\[ mX + m^3 - 2m = 0 \]
Factoring out \(m\):
\[ m(X + m^2 - 2) = 0 \]
This gives us one solution \(m = 0\) and the quadratic equation:
\[ m^2 + (X - 2) = 0 \]

### Step 5: Condition for real roots
For the quadratic \(m^2 + (X - 2) = 0\) to have real roots, the discriminant must be non-negative:
\[ X - 2 \geq 0 \]
Thus:
\[ X \geq 2 \]
Recalling that \(X = x - 1\), we substitute back:
\[ x - 1 \geq 2 \implies x \geq 3 \]

### Conclusion
The set of points on the x-axis from which all three normals to the parabola are real is:
\[ \boxed{x \geq 3} \]