The point of intersection of the tangent to the parabola `y^(2)=4x` which also touches `x^(2)+y^(2)=(1)/(2)` is

Options

A. `(-1, 0)`
B. `(-(1)/(2), 0)`
C. `(-2, 0)`
D. `((-3)/(2), 0)`

Answer

To find the point of intersection of the tangents to the parabola \( y^2 = 4x \) that also touch the circle \( x^2 + y^2 = \frac{1}{2} \), we can follow these steps:

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

### Step 2: Write the equation of the tangent to the circle
The equation of the tangent to the circle \( x^2 + y^2 = \frac{1}{2} \) at a point can be expressed as:
\[ y = mx \pm \frac{1}{\sqrt{2}} \sqrt{1 + m^2} \]
Here, \( r = \frac{1}{\sqrt{2}} \) is the radius of the circle.

### Step 3: Set the two equations equal
Since the tangent to the parabola also touches the circle, we can equate the two expressions for \( y \):
\[ mx + \frac{1}{m} = mx + \frac{1}{\sqrt{2}} \sqrt{1 + m^2} \]
Cancelling \( mx \) from both sides gives:
\[ \frac{1}{m} = \frac{1}{\sqrt{2}} \sqrt{1 + m^2} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sqrt{2} = m \sqrt{1 + m^2} \]
Squaring both sides results in:
\[ 2 = m^2 (1 + m^2) \]
This simplifies to:
\[ m^4 + m^2 - 2 = 0 \]

### Step 5: Solve the quadratic equation
Let \( u = m^2 \). The equation becomes:
\[ u^2 + u - 2 = 0 \]
Factoring gives:
\[ (u - 1)(u + 2) = 0 \]
Thus, \( u = 1 \) or \( u = -2 \). Since \( m^2 \) cannot be negative, we take \( m^2 = 1 \), which gives \( m = 1 \) or \( m = -1 \).

### Step 6: Find the equations of the tangents
For \( m = 1 \):
\[ y = x + 1 \]
For \( m = -1 \):
\[ y = -x - 1 \]

### Step 7: Find the intersection of the two tangents
To find the intersection point, set the two equations equal:
\[ x + 1 = -x - 1 \]
Solving this gives:
\[ 2x = -2 \implies x = -1 \]
Substituting \( x = -1 \) into one of the tangent equations (let's use \( y = x + 1 \)):
\[ y = -1 + 1 = 0 \]

### Conclusion
Thus, the point of intersection of the tangents is:
\[ (-1, 0) \]