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 \( P \) inside the parabola, we can follow these steps:

### Step 1: Understand the Parabola
The equation \( y^2 = 16x \) represents a parabola that opens to the right. The vertex of the parabola is at the origin (0, 0), and the focus is at the point (4, 0) since \( 4a = 16 \) gives \( a = 4 \).

**Hint:** Identify the characteristics of the parabola, including its vertex and focus.

### Step 2: Identify the Variable Point \( P \)
Let \( P(m, n) \) be a variable point inside the parabola. Since \( P \) lies inside the parabola, it must satisfy the inequality \( n^2 < 16m \).

**Hint:** Remember that the point \( P \) must be within the bounds set by the parabola's equation.

### Step 3: Set Up the Distance Equation
The points \( (x, y) \) on the parabola are equidistant from the point \( P(m, n) \). The distance from \( P \) to a point on the parabola is given by:
\[ d = \sqrt{(x - m)^2 + (y - n)^2} \]
For two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on the parabola to be equidistant from \( P \), we set the distances equal:
\[ \sqrt{(x_1 - m)^2 + (y_1 - n)^2} = \sqrt{(x_2 - m)^2 + (y_2 - n)^2} \]

**Hint:** Use the distance formula to establish the relationship between points on the parabola and the variable point \( P \).

### Step 4: Substitute the Parabola Equation
Since \( y^2 = 16x \), we can express \( x \) in terms of \( y \):
\[ x = \frac{y^2}{16} \]
Substituting this into the distance equation will yield a polynomial equation in terms of \( y \).

**Hint:** Substitute the parabola's equation into the distance formula to simplify the problem.

### Step 5: Analyze the Resulting Polynomial
The resulting polynomial equation will be of degree 4 (since it will involve \( y^4 \) when substituting \( x \)). A polynomial of degree 4 can have at most 4 real roots.

**Hint:** The degree of the polynomial indicates the maximum number of solutions (points) that can be found.

### Step 6: 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.

**Final Answer:** The maximum number of points is **4**.