The straight line `y=mx+c` is a focal length chord of parabola `y^2=4x` , which also touches the hyperbola `x^2-y^2=4` , then the value of `m` is

Options

A. `m=+-2/sqrt3`
B. `m=+-sqrt3/2`
C. `m=+-2/3`
D. `m=+-3/2`

Answer

To solve the problem, we need to find the value of \( m \) for the line \( y = mx + c \) that is a focal length chord of the parabola \( y^2 = 4x \) and also touches the hyperbola \( x^2 - y^2 = 4 \).

### Step-by-Step Solution:

1. **Identify the Parameters of the Parabola**:
The equation of the parabola is given as \( y^2 = 4x \). This can be rewritten in the standard form \( y^2 = 4ax \), where \( a = 1 \). The focus of this parabola is at the point \( (1, 0) \).

**Hint**: Remember that for a parabola in the form \( y^2 = 4ax \), the focus is located at \( (a, 0) \).

2. **Determine the Value of \( c \)**:
Since the line \( y = mx + c \) is a focal length chord, it passes through the focus of the parabola. Therefore, substituting the coordinates of the focus \( (1, 0) \) into the line equation gives:
\[ 0 = m(1) + c \implies c = -m \]

**Hint**: A focal length chord passes through the focus of the parabola.

3. **Substitute \( c \) into the Line Equation**:
The line can now be expressed as:
\[ y = mx - m \]

4. **Condition for Tangency with the Hyperbola**:
The hyperbola is given by \( x^2 - y^2 = 4 \). To find the condition for the line to touch this hyperbola, we substitute \( y = mx - m \) into the hyperbola equation:
\[ x^2 - (mx - m)^2 = 4 \]
Expanding this:
\[ x^2 - (m^2x^2 - 2mxm + m^2) = 4 \]
Simplifying gives:
\[ x^2 - m^2x^2 + 2m^2x - m^2 - 4 = 0 \]
This can be rearranged to:
\[ (1 - m^2)x^2 + 2m^2x - (m^2 + 4) = 0 \]

5. **Discriminant Condition for Tangency**:
For the line to touch the hyperbola, the discriminant of this quadratic equation must be zero:
\[ B^2 - 4AC = 0 \]
Here, \( A = 1 - m^2 \), \( B = 2m^2 \), and \( C = -(m^2 + 4) \).
Thus, we have:
\[ (2m^2)^2 - 4(1 - m^2)(-m^2 - 4) = 0 \]
Expanding this:
\[ 4m^4 + 4(1 - m^2)(m^2 + 4) = 0 \]
Simplifying gives:
\[ 4m^4 + 4(m^2 + 4 - m^4 - 4m^2) = 0 \]
Which simplifies to:
\[ 4m^4 - 12m^2 + 16 = 0 \]
Dividing through by 4:
\[ m^4 - 3m^2 + 4 = 0 \]

6. **Letting \( u = m^2 \)**:
Substitute \( u = m^2 \):
\[ u^2 - 3u + 4 = 0 \]
The discriminant of this quadratic is:
\[ (-3)^2 - 4(1)(4) = 9 - 16 = -7 \]
Since the discriminant is negative, we need to check our earlier steps for any errors.

7. **Revisiting the Tangent Condition**:
The correct condition for the line to touch the hyperbola can be derived from:
\[ c^2 = a^2m^2 - b^2 \]
where \( a^2 = 4 \) and \( b^2 = 4 \). Thus:
\[ (-m)^2 = 4m^2 - 4 \implies m^2 = 4m^2 - 4 \implies 3m^2 = 4 \implies m^2 = \frac{4}{3} \]
Taking the square root gives:
\[ m = \pm \frac{2\sqrt{3}}{3} \]

### Final Answer:
The value of \( m \) is \( \pm \frac{2\sqrt{3}}{3} \).