Equation of common tangent of parabola `y ^(2) = 8x and x ^(2) + y =0` is

Options

A. `y = 2x +1`
B. `x = y +1`
C. `2x - y + 1 =0`
D. `x + 2y + 1=0`

Answer

To find the equation of the common tangent to the parabolas \( y^2 = 8x \) and \( x^2 + y = 0 \), we can follow these steps:

### Step 1: Identify the equations of the parabolas
The first parabola is given by:
\[ y^2 = 8x \]
This can be rewritten in standard form as:
\[ y^2 = 4px \]
where \( p = 2 \).

The second parabola is given by:
\[ x^2 + y = 0 \]
This can be rewritten as:
\[ y = -x^2 \]
This parabola opens downwards.

### Step 2: Write the equation of the tangent to the first parabola
The equation of the tangent to the parabola \( y^2 = 8x \) can be expressed as:
\[ y = mx + \frac{4}{m} \]
where \( m \) is the slope of the tangent.

### Step 3: Write the equation of the tangent to the second parabola
The equation of the tangent to the parabola \( y = -x^2 \) can be expressed as:
\[ y = mx - \frac{1}{4m^2} \]
where \( m \) is the slope of the tangent.

### Step 4: Set the equations of the tangents equal to each other
To find the common tangent, we set the two equations equal to each other:
\[ mx + \frac{4}{m} = mx - \frac{1}{4m^2} \]

### Step 5: Simplify the equation
Since \( mx \) cancels out, we have:
\[ \frac{4}{m} = -\frac{1}{4m^2} \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 4 \cdot 4m^2 = -1 \cdot m \]
This simplifies to:
\[ 16m^2 + m = 0 \]

### Step 7: Factor the equation
Factoring out \( m \) gives:
\[ m(16m + 1) = 0 \]

### Step 8: Solve for \( m \)
Setting each factor to zero gives:
1. \( m = 0 \)
2. \( 16m + 1 = 0 \) which gives \( m = -\frac{1}{16} \)

### Step 9: Find the equations of the common tangents
For \( m = 0 \):
\[ y = 0x + \frac{4}{0} \] (not valid)

For \( m = -\frac{1}{16} \):
Substituting into the tangent equation for \( y^2 = 8x \):
\[ y = -\frac{1}{16}x + \frac{4}{-\frac{1}{16}} = -\frac{1}{16}x - 64 \]

### Step 10: Convert to standard form
To convert this into standard form:
\[ y + \frac{1}{16}x + 64 = 0 \]
Multiplying through by 16 to eliminate the fraction gives:
\[ 16y + x + 1024 = 0 \]

### Final Answer
The equation of the common tangent is:
\[ x + 16y + 1024 = 0 \]

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