Equation of tangent of a conjugate hyperbola in slope form with a slope m inclined to positive x axis

To find the equation of the tangent to a conjugate hyperbola in slope form, we first need to understand the general characteristics of conjugate hyperbolas and how tangents are derived from them. A conjugate hyperbola is typically represented by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \). The tangent line to this hyperbola at a point can be expressed in terms of its slope.

Understanding the Conjugate Hyperbola

The standard form of a conjugate hyperbola is given by:

• For the hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \)
• For the conjugate hyperbola: \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = -1 \)

In this case, the conjugate hyperbola opens vertically, and its asymptotes are given by the lines \( y = \pm \frac{b}{a} x \).

Equation of the Tangent Line

To derive the equation of the tangent line with slope \( m \), we can use the point-slope form of a line. The general equation of the tangent to the conjugate hyperbola can be expressed as:

\( y = mx + c \)

where \( c \) is the y-intercept. To find \( c \), we need to ensure that this line is tangent to the hyperbola. This means that the line must intersect the hyperbola at exactly one point.

Finding the Condition for Tangency

Substituting \( y = mx + c \) into the equation of the conjugate hyperbola gives:

\( \frac{(mx + c)^2}{b^2} - \frac{x^2}{a^2} = -1 \)

Expanding this, we get:

\( \frac{m^2x^2 + 2mcx + c^2}{b^2} - \frac{x^2}{a^2} + 1 = 0 \)

Rearranging leads to a quadratic equation in \( x \):

\( \left(\frac{m^2}{b^2} - \frac{1}{a^2}\right)x^2 + \frac{2mc}{b^2}x + \left(\frac{c^2}{b^2} + 1\right) = 0 \)

For this quadratic to have exactly one solution (indicating tangency), the discriminant must be zero:

\( \Delta = \left(\frac{2mc}{b^2}\right)^2 - 4\left(\frac{m^2}{b^2} - \frac{1}{a^2}\right)\left(\frac{c^2}{b^2} + 1\right) = 0 \)

Solving for the y-intercept \( c \)

By solving the discriminant equation, we can find the value of \( c \) in terms of \( m \), \( a \), and \( b \). This will give us the specific y-intercept for the tangent line with slope \( m \).

Final Tangent Equation

Once we have determined \( c \), we can substitute it back into the line equation \( y = mx + c \). This gives us the complete equation of the tangent line to the conjugate hyperbola in slope form.

In summary, the process involves understanding the hyperbola's properties, substituting the line equation into the hyperbola's equation, and ensuring the discriminant condition for tangency is satisfied. This method provides a systematic approach to finding the tangent line with a given slope to a conjugate hyperbola.