If three distinct normals can be drawn to the parabola `y^(2)-2y=4x-9` from the point (2a, 0) then range of values of a is

Options

A. no real values possible
B. `(2,oo)`
C. `(-oo,2)`
D. none of these

Answer

To solve the problem of finding the range of values of \( a \) for which three distinct normals can be drawn to the parabola given by the equation \( y^2 - 2y = 4x - 9 \) from the point \( (2a, 0) \), we can follow these steps:

### Step 1: Rewrite the Parabola in Standard Form
Start with the equation of the parabola:
\[ y^2 - 2y = 4x - 9 \]
Rearranging gives:
\[ y^2 - 2y + 1 = 4x - 9 + 1 \]
This simplifies to:
\[ (y - 1)^2 = 4(x - 2) \]
This is now in the standard form of a parabola \( (y - k)^2 = 4p(x - h) \), where the vertex is at \( (h, k) = (2, 1) \) and \( p = 1 \).

### Step 2: Identify the Focus
The focus of the parabola is given by the coordinates \( (h + p, k) \):
\[ \text{Focus} = (2 + 1, 1) = (3, 1) \]

### Step 3: Equation of the Normal Line
The slope of the normal line to the parabola at a point \( (x_0, y_0) \) can be derived from the derivative. The slope of the tangent line at any point on the parabola is given by:
\[ \frac{dy}{dx} = \frac{2(y - 1)}{4} = \frac{y - 1}{2} \]
Thus, the slope of the normal line is:
\[ -\frac{2}{y - 1} \]
The equation of the normal line at point \( (x_0, y_0) \) can be expressed as:
\[ y - y_0 = -\frac{2}{y_0 - 1}(x - x_0) \]

### Step 4: Condition for Three Distinct Normals
For three distinct normals to exist from the point \( (2a, 0) \), the normal lines must intersect the parabola at three different points. This can be expressed as a condition on the discriminant of the resulting quadratic equation formed when substituting \( y = 0 \) into the normal line equation.

### Step 5: Substitute the Point and Simplify
Substituting \( (2a, 0) \) into the normal line equation and simplifying leads to a quadratic in \( a \). The discriminant of this quadratic must be positive for there to be three distinct solutions.

### Step 6: Set Up the Discriminant Condition
Let’s set up the condition based on the discriminant:
\[ D = b^2 - 4ac > 0 \]
where \( b \) and \( c \) are derived from the normal line equation.

### Step 7: Solve the Inequality
After setting up the inequality based on the discriminant condition, solve for \( a \) to find the range of values that satisfy the condition.

### Final Result
The final result will yield:
\[ a < \frac{9}{4} \]