Find the condition that the straight line `y = mx + c ` touches the hyperbola `x^(2) - y^(2) = a^(2) `.

Answer

To find the condition that the straight line \( y = mx + c \) touches the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Write the equation of the hyperbola in standard form
The hyperbola is given as:
\[ x^2 - y^2 = a^2 \]
This can be rewritten in standard form as:
\[ \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \]
Here, we identify \( b^2 = a^2 \), so \( b = a \).

### Step 2: Write the equation of the tangent to the hyperbola
The equation of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) at the point where it touches is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Substituting \( b^2 = a^2 \):
\[ y = mx \pm \sqrt{a^2 m^2 - a^2} \]
This simplifies to:
\[ y = mx \pm a\sqrt{m^2 - 1} \]

### Step 3: Set the equations equal for tangency condition
For the line \( y = mx + c \) to touch the hyperbola, it must equal the tangent line at some point. Thus, we set:
\[ mx + c = mx \pm a\sqrt{m^2 - 1} \]
This leads to:
\[ c = \pm a\sqrt{m^2 - 1} \]

### Step 4: Determine the condition for the line to be tangent
The line \( y = mx + c \) will touch the hyperbola if the y-intercept \( c \) satisfies:
\[ c = a\sqrt{m^2 - 1} \quad \text{or} \quad c = -a\sqrt{m^2 - 1} \]
This means that for the line to be tangent to the hyperbola, the value of \( c \) must be equal to \( \pm a\sqrt{m^2 - 1} \).

### Final Condition
Thus, the condition that the line \( y = mx + c \) touches the hyperbola \( x^2 - y^2 = a^2 \) is:
\[ c^2 = a^2(m^2 - 1) \]