Equation of a common tangent to the parabola `y^(2)=4x` and the hyperbola xy=2 is

Options

A. x+2y+4=0
B. x-2y+4=0
C. x+y+1=0
D. 4x+2y+1=0

Answer

To find the equation of a common tangent to the parabola \( y^2 = 4x \) and the hyperbola \( xy = 2 \), we can follow these steps:

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

**Hint:** Remember that the general form of the tangent line to a parabola can be derived using the point-slope form of a line.

### Step 2: Substitute the tangent equation into the hyperbola
Next, we substitute \( y = mx + \frac{1}{m} \) into the hyperbola equation \( xy = 2 \):
\[ x \left( mx + \frac{1}{m} \right) = 2 \]
This simplifies to:
\[ mx^2 + x \cdot \frac{1}{m} - 2 = 0 \]

**Hint:** Make sure to distribute \( x \) correctly when substituting into the hyperbola equation.

### Step 3: Rearrange the equation
Rearranging gives us:
\[ mx^2 + \frac{1}{m}x - 2 = 0 \]

**Hint:** Identify the coefficients \( a = m \), \( b = \frac{1}{m} \), and \( c = -2 \) for the quadratic equation.

### Step 4: Use the condition for common tangents
For the tangent to be common to both curves, the quadratic must have equal roots. This occurs when the discriminant is zero:
\[ b^2 - 4ac = 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \left(\frac{1}{m}\right)^2 - 4(m)(-2) = 0 \]
This simplifies to:
\[ \frac{1}{m^2} + 8m = 0 \]

**Hint:** Setting the discriminant to zero ensures that the quadratic has a double root, which corresponds to the common tangent.

### Step 5: Solve for \( m \)
Multiplying through by \( m^2 \) (assuming \( m \neq 0 \)):
\[ 1 + 8m^3 = 0 \]
Thus,
\[ 8m^3 = -1 \quad \Rightarrow \quad m^3 = -\frac{1}{8} \quad \Rightarrow \quad m = -\frac{1}{2} \]

**Hint:** When solving for \( m \), remember to take the cube root correctly.

### Step 6: Substitute \( m \) back into the tangent equation
Now, substituting \( m = -\frac{1}{2} \) back into the tangent equation:
\[ y = -\frac{1}{2}x + \frac{1}{-\frac{1}{2}} = -\frac{1}{2}x - 2 \]

**Hint:** Ensure you correctly substitute and simplify the expression for \( y \).

### Step 7: Rearranging the tangent equation
Rearranging gives:
\[ y + \frac{1}{2}x + 2 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2y + x + 4 = 0 \]
or
\[ x - 2y + 4 = 0 \]

**Hint:** Rearranging to standard form can help clarify the final equation of the tangent.

### Final Answer
The equation of the common tangent to the parabola \( y^2 = 4x \) and the hyperbola \( xy = 2 \) is:
\[ x - 2y + 4 = 0 \]