Straight line `Ax+By+D=0` would be tangent to `xy=c^2`, if

Options

A. `A gt 0 , B gt 0 `
B. `A lt 0, B lt 0 `
C. `A gt 0, B lt 0 `
D. `A lt 0, B gt 0 `

Answer

To determine the conditions under which the straight line \( Ax + By + D = 0 \) is tangent to the hyperbola defined by \( xy = c^2 \), we can follow these steps:

### Step 1: Understand the Equation of the Hyperbola
The equation \( xy = c^2 \) represents a hyperbola. We need to find the conditions for the line to be tangent to this hyperbola.

### Step 2: Differentiate the Hyperbola
To find the slope of the tangent line at any point on the hyperbola, we differentiate the equation \( xy = c^2 \):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(c^2) \]
Using the product rule, we get:
\[ y + x \frac{dy}{dx} = 0 \]
From this, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 3: Find the Equation of the Tangent Line
At a point \( (x_1, y_1) \) on the hyperbola, the slope of the tangent line is:
\[ m = -\frac{y_1}{x_1} \]
The equation of the tangent line at this point can be expressed as:
\[ y - y_1 = m(x - x_1) \]
Substituting the slope \( m \):
\[ y - y_1 = -\frac{y_1}{x_1}(x - x_1) \]
Rearranging gives:
\[ y x_1 + y_1 x = 2y_1 x_1 \]

### Step 4: Rearranging the Tangent Equation
Rearranging the equation yields:
\[ yx_1 + y_1x = 2y_1x_1 \]
This can be rewritten as:
\[ y = \frac{2y_1x_1 - y_1x}{x_1} \]

### Step 5: Compare with the Given Line Equation
The given line is \( Ax + By + D = 0 \). We can express \( y \) in terms of \( x \):
\[ y = -\frac{A}{B}x - \frac{D}{B} \]
Now we compare the coefficients from both equations:
1. The slope \( -\frac{A}{B} \) should equal \( -\frac{y_1}{x_1} \).
2. The constant term \( -\frac{D}{B} \) should equal \( 2y_1 \).

### Step 6: Analyze the Conditions
From the slope comparison:
\[ \frac{A}{B} = \frac{y_1}{x_1} \]
Since \( x_1y_1 = c^2 > 0 \) (as \( c^2 \) is positive), both \( x_1 \) and \( y_1 \) must have the same sign. This leads to two cases:
1. \( A > 0 \) and \( B > 0 \)
2. \( A < 0 \) and \( B < 0 \)

### Conclusion
The straight line \( Ax + By + D = 0 \) will be tangent to the hyperbola \( xy = c^2 \) if:
- \( A > 0 \) and \( B > 0 \) (both coefficients positive)
- or \( A < 0 \) and \( B < 0 \) (both coefficients negative)