Set of values of 'h' for which the number of distinct common normals of `(x-2)^(2)=4(y-3)` and
`x^(2)+y^(2)-2x-hy-c=0(cgt0)" is 3, is"`

Options

A. `(2,oo)`
B. `(4,oo)`
C. `(2,4)`
D. `(10,oo)`

Answer

To solve the problem of finding the set of values of \( h \) for which the number of distinct common normals of the parabola \( (x-2)^2 = 4(y-3) \) and the circle given by \( x^2 + y^2 - 2x - hy - c = 0 \) (where \( c > 0 \)) is 3, we can follow these steps:

### Step 1: Identify the Parabola and Circle Equations
The parabola can be rewritten as:
\[ y = \frac{1}{4}(x-2)^2 + 3 \]
The circle equation can be rearranged to:
\[ x^2 + y^2 - 2x - hy - c = 0 \]

### Step 2: Convert Circle Equation to Standard Form
To convert the circle equation into standard form, we complete the square:
\[ (x-1)^2 + \left(y - \frac{h}{2}\right)^2 = c + 1 - \frac{h^2}{4} \]
This shows that the center of the circle is \( (1, \frac{h}{2}) \) and the radius is \( R = \sqrt{c + 1 - \frac{h^2}{4}} \).

### Step 3: Find the Normal to the Parabola
The equation of the normal to the parabola at a point \( (x_0, y_0) \) can be written as:
\[ y - y_0 = -\frac{1}{m}(x - x_0) \]
where \( m \) is the slope of the tangent at that point. For the parabola, the slope of the tangent at any point can be derived from its derivative.

### Step 4: Set Up the Condition for Common Normals
For the parabola and the circle to have common normals, the normals must intersect the circle at two distinct points. This leads to a cubic equation in terms of the slope \( m \).

### Step 5: Analyze the Cubic Equation
The cubic equation derived from the conditions of the normals will have three distinct real roots if its discriminant is positive. The cubic can be expressed as:
\[ f(m) = 2m^3 + (10 - h)m - 2 = 0 \]
To ensure three distinct real roots, we need to analyze the discriminant of this cubic equation.

### Step 6: Calculate the Condition for Distinct Roots
The condition for the cubic \( f(m) \) to have three distinct roots is given by:
\[ \Delta > 0 \]
For a cubic equation \( ax^3 + bx^2 + cx + d = 0 \), the discriminant can be calculated, but for simplicity, we can also analyze the derivative \( f'(m) \) to find critical points.

### Step 7: Find Critical Points
The derivative is:
\[ f'(m) = 6m^2 + (10 - h) \]
Setting \( f'(m) = 0 \) gives:
\[ 6m^2 + (10 - h) = 0 \implies m^2 = \frac{h - 10}{6} \]
For \( m^2 \) to be non-negative, we require:
\[ h - 10 \geq 0 \implies h \geq 10 \]

### Step 8: Determine the Range of \( h \)
Since we need \( c > 0 \) and the circle's radius must be positive, we also need:
\[ c + 1 - \frac{h^2}{4} > 0 \implies c > \frac{h^2}{4} - 1 \]
This gives us a condition on \( h \) as well.

### Conclusion
Combining the conditions, we find that for the number of distinct common normals to be 3, the values of \( h \) must satisfy:
\[ h \in [10, \infty) \]

### Final Answer
The set of values of \( h \) for which the number of distinct common normals is 3 is:
\[ \boxed{[10, \infty)} \]