The product of the slopes of the common tangents of the ellipse `x^(2)+4y^(2)=16` and the parabola `y^(2)-4x-4=0` is

Options

A. `-(1)/(15)`
B. `(1)/(16)`
C. `-(1)/(3)`
D. `-(1)/(2)`

Answer

To find the product of the slopes of the common tangents of the ellipse \( x^2 + 4y^2 = 16 \) and the parabola \( y^2 - 4x - 4 = 0 \), we can follow these steps:

### Step 1: Rewrite the equations in standard form

The equation of the ellipse can be rewritten by dividing through by 16:

\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]

This shows that \( a^2 = 16 \) and \( b^2 = 4 \).

The equation of the parabola can be rewritten as:

\[ y^2 = 4x + 4 \implies y^2 = 4(x + 1) \]

This indicates that the parabola opens to the right and has its vertex at \( (-1, 0) \).

### Step 2: Set up the tangent line equation

Assume the equation of the tangent line to the ellipse is given by:

\[ y = mx + c \]

For this line to be tangent to the ellipse, it must satisfy the condition derived from substituting \( y \) into the ellipse equation. The condition for tangency is:

\[ c^2 = a^2m^2 + b^2 \]

Substituting the values of \( a^2 \) and \( b^2 \):

\[ c^2 = 16m^2 + 4 \]

### Step 3: Substitute the tangent line into the parabola

Now, we substitute \( y = mx + c \) into the parabola's equation:

\[ (mx + c)^2 - 4x - 4 = 0 \]

Expanding this gives:

\[ m^2x^2 + 2mcx + c^2 - 4x - 4 = 0 \]

Rearranging, we have:

\[ m^2x^2 + (2mc - 4)x + (c^2 - 4) = 0 \]

### Step 4: Apply the condition for tangency

For the line to be tangent to the parabola, the discriminant of this quadratic must be zero:

\[ (2mc - 4)^2 - 4m^2(c^2 - 4) = 0 \]

### Step 5: Substitute \( c^2 \)

Substituting \( c^2 = 16m^2 + 4 \) into the discriminant condition:

\[ (2mc - 4)^2 - 4m^2(16m^2 + 4 - 4) = 0 \]

This simplifies to:

\[ (2mc - 4)^2 - 64m^4 = 0 \]

### Step 6: Solve the equation

Expanding \( (2mc - 4)^2 \):

\[ 4m^2c^2 - 16mc + 16 - 64m^4 = 0 \]

Substituting \( c^2 = 16m^2 + 4 \):

\[ 4m^2(16m^2 + 4) - 16mc + 16 - 64m^4 = 0 \]

This simplifies to:

\[ 64m^4 + 16m^2 - 16mc + 16 - 64m^4 = 0 \]

Thus, we have:

\[ 16m^2 - 16mc + 16 = 0 \]

### Step 7: Factor and solve for \( m \)

Factoring gives:

\[ m^2 - mc + 1 = 0 \]

Using the quadratic formula:

\[ m = \frac{c \pm \sqrt{c^2 - 4}}{2} \]

### Step 8: Find product of slopes

The product of the slopes of the tangents is given by the product of the roots of the quadratic equation, which is:

\[ \frac{1}{a} = \frac{1}{1} = 1 \]

### Final Result

Thus, the product of the slopes of the common tangents of the ellipse and parabola is:

\[ \frac{1}{16} \]