Equation of a common tangent to the circle `x^2+y^2=16`, parabola `x^2=y-4` and the ellipse `x^2/25+y^2/16=0` is

Options

A. x=4
B. x=-4
C. y=4
D. y=5

Answer

To find the equation of a common tangent to the circle \(x^2 + y^2 = 16\), the parabola \(x^2 = y - 4\), and the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), we can follow these steps:

### Step 1: Identify the curves
1. **Circle**: The equation \(x^2 + y^2 = 16\) represents a circle centered at the origin (0,0) with a radius of 4.
2. **Parabola**: The equation \(x^2 = y - 4\) can be rewritten as \(y = x^2 + 4\). This parabola opens upwards and has its vertex at (0, 4).
3. **Ellipse**: The equation \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) represents an ellipse centered at the origin with semi-major axis 5 (along the x-axis) and semi-minor axis 4 (along the y-axis).

### Step 2: Visualize the curves
- The circle has points at (4,0), (0,4), (-4,0), and (0,-4).
- The parabola passes through (0,4) and opens upwards.
- The ellipse has points at (5,0), (0,4), (-5,0), and (0,-4).

### Step 3: Determine the common tangent
To find the common tangent, we can analyze the geometry of the curves:
- The circle and the parabola touch at the point (0,4).
- The ellipse also touches the line \(y = 4\) at the point (0,4).

### Step 4: Write the equation of the tangent line
Since all three curves touch the line \(y = 4\), this line serves as a common tangent to all three curves.

### Conclusion
Thus, the equation of the common tangent to the given circle, parabola, and ellipse is:
\[ \boxed{y = 4} \]