Find the equations of the common tangents to the circle `x^2+y^2 = 8` and the parabola `y^2 = 16x.`

Options

A. `x=y+4`
B. `y=x-4`
C. `y=x+4`
D. `x=y-4`

Answer

To find the equations of the common tangents to the circle \(x^2 + y^2 = 8\) and the parabola \(y^2 = 16x\), we will follow these steps:

### Step 1: Identify the parameters of the circle and parabola
The equation of the circle is given by:
\[ x^2 + y^2 = 8 \]
This can be rewritten in standard form as:
\[ x^2 + y^2 = r^2 \quad \text{where } r = 2\sqrt{2} \]

The equation of the parabola is given by:
\[ y^2 = 16x \]
This can be rewritten in standard form as:
\[ y^2 = 4ax \quad \text{where } 4a = 16 \Rightarrow a = 4 \]

### Step 2: Write the equation of the tangent to the circle
The equation of the tangent to the circle in slope form is:
\[ y = mx \pm r\sqrt{1 + m^2} \]
Substituting \(r = 2\sqrt{2}\):
\[ y = mx \pm 2\sqrt{2}\sqrt{1 + m^2} \]

### Step 3: Write the equation of the tangent to the parabola
The equation of the tangent to the parabola in slope form is:
\[ y = mx + \frac{4}{m} \]

### Step 4: Set the two tangent equations equal
For the tangents to be common, the constant terms must be equal. Therefore, we equate:
\[ \frac{4}{m} = \pm 2\sqrt{2}\sqrt{1 + m^2} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{4}{m}\right)^2 = (2\sqrt{2})^2(1 + m^2) \]
This simplifies to:
\[ \frac{16}{m^2} = 8(1 + m^2) \]
Multiplying through by \(m^2\) to eliminate the fraction yields:
\[ 16 = 8m^2 + 8m^4 \]
Rearranging gives:
\[ 8m^4 + 8m^2 - 16 = 0 \]
Dividing the entire equation by 8:
\[ m^4 + m^2 - 2 = 0 \]

### Step 6: Let \(u = m^2\) and solve the quadratic
Let \(u = m^2\), then we have:
\[ u^2 + u - 2 = 0 \]
Factoring gives:
\[ (u - 1)(u + 2) = 0 \]
Thus, \(u = 1\) or \(u = -2\). Since \(u = m^2\) cannot be negative, we have:
\[ m^2 = 1 \Rightarrow m = \pm 1 \]

### Step 7: Find the equations of the tangents
Substituting \(m = 1\) into the tangent equation for the parabola:
\[ y = x + \frac{4}{1} = x + 4 \]
Substituting \(m = -1\):
\[ y = -x + \frac{4}{-1} = -x - 4 \]

### Final Result
The equations of the common tangents are:
\[ y = x + 4 \quad \text{and} \quad y = -x - 4 \]