The condition that the line x=my+c may be a tangent of
`x^(2)/a^(2)-y^(2)/b^(2)=-1` is

Options

A. `c^(2)=a^(2)m^(2)-b^(2)`
B. `c^(2)=a^(2)-b^(2)m^2`
C. `c^(2)=b^(2)-a^(2)m^(2)`
D. `c^(2)=b^(2)m^(2)-a^(2)`

Answer

To determine the condition under which the line \( x = my + c \) is a tangent to the hyperbola given by

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1, \]

we can follow these steps:

### Step 1: Rewrite the Line Equation
First, we rewrite the line equation \( x = my + c \) in standard form:

\[ x - my - c = 0. \]

Let this be equation (3).

### Step 2: Use the Tangent Equation of the Hyperbola
The standard form of the tangent to the hyperbola at a point \( (x_1, y_1) \) is given by:

\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1. \]

Let’s denote this as equation (2).

### Step 3: Establish the Condition for Tangency
For the line to be a tangent to the hyperbola, equations (2) and (3) must represent the same line. Therefore, we can equate the coefficients of \( x \), \( y \), and the constant term from both equations.

### Step 4: Coefficient Comparison
From equation (2), we have:

- Coefficient of \( x \): \( \frac{x_1}{a^2} \)
- Coefficient of \( y \): \( -\frac{y_1}{b^2} \)
- Constant term: \( -1 \)

From equation (3), we have:

- Coefficient of \( x \): \( 1 \)
- Coefficient of \( y \): \( -m \)
- Constant term: \( -c \)

Setting up the ratios:

\[ \frac{x_1}{a^2} = 1, \quad -\frac{y_1}{b^2} = -m, \quad -1 = -c. \]

### Step 5: Solve for \( x_1 \) and \( y_1 \)
From the first equation, we get:

\[ x_1 = a^2. \]

From the second equation, we have:

\[ y_1 = mb^2. \]

From the third equation, we find:

\[ c = 1. \]

### Step 6: Substitute \( x_1 \) and \( y_1 \) into the Line Equation
Now, substituting \( x_1 \) and \( y_1 \) back into the line equation \( x - my - c = 0 \):

\[ a^2 - m(mb^2) - 1 = 0. \]

This simplifies to:

\[ a^2 - m^2b^2 - 1 = 0. \]

### Step 7: Rearranging the Equation
Rearranging gives us the condition:

\[ c^2 = a^2 - m^2b^2. \]

This is the required condition for the line \( x = my + c \) to be a tangent to the hyperbola.

### Final Condition
Thus, the condition that the line \( x = my + c \) may be a tangent to the hyperbola is:

\[ c^2 = a^2 - m^2b^2. \]

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