The maximum number of points on the parabola `y^(2)=16x` which re equidistant from a variable point P (which lie inside the parabola) are

Answer

To solve the problem of finding the maximum number of points on the parabola \( y^2 = 16x \) that are equidistant from a variable point \( P \) (which lies inside the parabola), we can follow these steps:

### Step 1: Understand the Parabola and the Circle
The given parabola is \( y^2 = 16x \). This can be rewritten in standard form as \( x = \frac{y^2}{16} \). The points on this parabola can be represented as \( (x, y) \) where \( x \) is expressed in terms of \( y \).

The variable point \( P \) can be represented as \( (m, n) \), where \( m \) and \( n \) are coordinates of point \( P \) inside the parabola.

### Step 2: Equation of the Circle
The points that are equidistant from point \( P \) will lie on a circle centered at \( P \) with radius \( R \). The equation of this circle is given by:
\[ (x - m)^2 + (y - n)^2 = R^2 \]

### Step 3: Substitute the Parabola Equation into the Circle Equation
To find the points of intersection between the parabola and the circle, we substitute \( x \) from the parabola equation into the circle equation:
\[ \left(\frac{y^2}{16} - m\right)^2 + (y - n)^2 = R^2 \]

### Step 4: Expand and Rearrange the Equation
Expanding the left-hand side:
\[ \left(\frac{y^2}{16} - m\right)^2 = \frac{y^4}{256} - \frac{my^2}{8} + m^2 \]
\[ (y - n)^2 = y^2 - 2ny + n^2 \]
Combining these, we have:
\[ \frac{y^4}{256} - \frac{my^2}{8} + m^2 + y^2 - 2ny + n^2 = R^2 \]

### Step 5: Collect Like Terms
Rearranging gives us a polynomial in \( y \):
\[ \frac{y^4}{256} + \left(1 - \frac{m}{8}\right)y^2 - 2ny + (m^2 + n^2 - R^2) = 0 \]

### Step 6: Determine the Degree of the Polynomial
This is a quartic equation (degree 4) in \( y \). The maximum number of real roots of a polynomial is equal to its degree. Therefore, this quartic equation can have up to 4 real roots.

### Conclusion
Thus, the maximum number of points on the parabola \( y^2 = 16x \) that can be equidistant from a variable point \( P \) inside the parabola is **4**.