Asymptotic Lines of Hyperbola and Conjugate Hyperbola

1. Every Hyperbola and it's corresponding Conjugate Hyperbola has a Pair of Asymptotic Lines associated with them. These Pair of Asymptotic Lines pass through the Center of the Hyperbola / Conjugate Hyperbola and are Parallel to the Lines that pass through a Vertex and a Covertex of the Hyperbolas.
2. The following gives the Derivation equations for the Pair of Asymptotic Lines associated with an \(X\)-Transpose Hyperbola having it's Center at Origin and it's corresponding Conjugate Hyperbola
   
   \(\frac{x^2}{a^2} - \frac{y^2}{b^2} =1 \)   ...(1)
   
   \(\frac{y^2}{b^2} - \frac{x^2}{a^2} =1 \)   ...(2)
   
   Equation (1) above gives the equation of an \(X\)-Transpose Hyperbola having it's Center at Origin, Length of Semi-Transpose Axis \(a\) and Length of Semi-Conjugate Axis \(b\).
   
   Equation (2) above gives the equation of an \(Y\)-Transpose Hyperbola having it's Center at Origin, Length of Semi-Transpose Axis \(b\) and Length of Semi-Conjugate Axis \(a\).
   
   The Hyperbolas given by equation (1) and (2) are Conjugates of each other.
   
   One of the Asymptotic Line to these Hyperbolas is Parallel to the Line Passing through the Coordinates \((0,b)\) and \((a,0)\) which has the Slope \({\Large \frac{-b}{a}}\) (or Parallel to the Line Passing through the Coordinates \((-a,0)\) and \((0,-b)\) which also has the Slope \({\Large \frac{-b}{a}}\) ). Therefore this Asymptotic Line also has a Slope of \({\Large \frac{-b}{a}}\), and since it passes through the Origin, it's equation can be derived as
   
   \(\frac{y-0}{x-0}= \frac{-b}{a}\hspace{3mm}\Rightarrow \frac{y}{x}= \frac{-b}{a}\hspace{3mm}\Rightarrow ay=-bx\hspace{3mm}\Rightarrow bx+ay=0\)   ...(3)
   
   The other Asymptotic Line to these Hyperbolas is Parallel to the Line Passing through the Coordinates \((0,b)\) and \((-a,0)\) which has the Slope \({\Large \frac{b}{a}}\) (or Parallel to the Line Passing through the Coordinates \((a,0)\) and \((0,-b)\) which also has the Slope \({\Large \frac{b}{a}}\) ). Therefore this Asymptotic Line also has a Slope of \({\Large \frac{b}{a}}\), and since it passes through the Origin, it's equation can be derived as
   
   \(\frac{y-0}{x-0}= \frac{b}{a}\hspace{3mm}\Rightarrow \frac{y}{x}= \frac{b}{a}\hspace{3mm}\Rightarrow ay=bx\hspace{3mm}\Rightarrow bx-ay=0\)   ...(4)
   
   The equations (3) and (4) above give the Equations of Asymptotic Lines corresponding to Hyperbolas given by equations (1) and (2) above.
3. The following gives the Derivation equations for the Pair of Asymptotic Lines associated with an \(Y\)-Transpose Hyperbola having it's Center at Origin and it's corresponding Conjugate Hyperbola
   
   \(\frac{y^2}{a^2} - \frac{x^2}{b^2} =1 \)   ...(5)
   
   \(\frac{x^2}{b^2} - \frac{y^2}{a^2} =1 \)   ...(6)
   
   Equation (5) above gives the equation of an \(Y\)-Transpose Hyperbola having it's Center at Origin, Length of Semi-Transpose Axis \(a\) and Length of Semi-Conjugate Axis \(b\).
   
   Equation (6) above gives the equation of an \(X\)-Transpose Hyperbola having it's Center at Origin, Length of Semi-Transpose Axis \(b\) and Length of Semi-Conjugate Axis \(a\).
   
   The Hyperbolas given by equation (5) and (6) are Conjugates of each other.
   
   One of the Asymptotic Line to these Hyperbolas is Parallel to the Line Passing through the Coordinates \((0,a)\) and \((b,0)\) which has the Slope \({\Large \frac{-a}{b}}\) (or Parallel to the Line Passing through the Coordinates \((-b,0)\) and \((0,-a)\) which also has the Slope \({\Large \frac{-a}{b}}\) ). Therefore this Asymptotic Line also has a Slope of \({\Large \frac{-a}{b}}\), and since it passes through the Origin, it's equation can be derived as
   
   \(\frac{y-0}{x-0}= \frac{-a}{b}\hspace{3mm}\Rightarrow \frac{y}{x}= \frac{-a}{b}\hspace{3mm}\Rightarrow by=-ax\hspace{3mm}\Rightarrow ax+by=0\)   ...(7)
   
   The other Asymptotic Line to these Hyperbolas is Parallel to the Line Passing through the Coordinates \((0,a)\) and \((-b,0)\) which has the Slope \({\Large \frac{a}{b}}\) (or Parallel to the Line Passing through the Coordinates \((b,0)\) and \((0,-a)\) which also has the Slope \({\Large \frac{a}{b}}\) ). Therefore this Asymptotic Line also has a Slope of \({\Large \frac{a}{b}}\), and since it passes through the Origin, it's equation can be derived as
   
   \(\frac{y-0}{x-0}= \frac{a}{b}\hspace{3mm}\Rightarrow \frac{y}{x}= \frac{a}{b}\hspace{3mm}\Rightarrow by=ax\hspace{3mm}\Rightarrow ax-by=0\)   ...(8)
   
   The equations (7) and (8) above give the Equations of Asymptotic Lines corresponding to Hyperbolas given by equations (5) and (6) above.
4. The following gives the Derivation equations for the Pair of Asymptotic Lines associated with any Rotated and Translated Hyperbola and it's corresponding Conjugate Hyperbola
   
   Rotating/Translating the Hyperbolas/Connjugate Hyperbolas also rotates / translates their corresponding Asymptotic Lines similarly.
   
   Thus Rotating the Hyperbolas given by equations (1) and (2) by a Counter Clockwise Angle \(\theta\) also Rotates the Asymptotic Line given by equation (3) Counter Clockwise by Angle \(\theta\) as follows
   
   \(b(x\cos\theta + y\sin\theta) + a(y\cos\theta - x\sin\theta) = 0\)
   
   \(\Rightarrow b\cos\theta x + b \sin\theta y + a\cos\theta y - a\sin\theta x = 0\)
   
   \(\Rightarrow (b\cos\theta - a\sin\theta) x + (a\cos\theta + b\sin\theta) y = 0\)   ...(9)
   
   Similarly, the Asymptotic Line given by equation (4) also gets rotated Counter Clockwise by Angle \(\theta\) as follows
   
   \(b(x\cos\theta + y\sin\theta) - a(y\cos\theta - x\sin\theta) = 0\)
   
   \(\Rightarrow b\cos\theta x + b \sin\theta y - a\cos\theta y + a\sin\theta x = 0\)
   
   \(\Rightarrow (b\cos\theta + a\sin\theta) x - (a\cos\theta - b\sin\theta) y = 0\)   ...(10)
   
   Now, Translating the Rotated Hyperbolas by an offset \((x_c,y_c)\) also Translates the Rotated Asymptotic Line given by equation (9) by an offset \((x_c,y_c)\) as follows
   
   \((b\cos\theta - a\sin\theta) (x-x_c) + (a\cos\theta + b\sin\theta) (y-y_c) = 0\)
   
   \(\Rightarrow (b\cos\theta - a\sin\theta) x + (a\cos\theta + b\sin\theta) y - (b\cos\theta - a\sin\theta)x_c - (a\cos\theta + b\sin\theta)y_c = 0\)   ...(11)
   
   Similarly, the Rotated Asymptotic Line given by equation (10) gets translated by an offset \((x_c,y_c)\) as follows
   
   \((b\cos\theta + a\sin\theta) (x-x_c) - (a\cos\theta - b\sin\theta) (y-y_c) = 0\)   ...(11)
   
   \(\Rightarrow (b\cos\theta + a\sin\theta) x - (a\cos\theta - b\sin\theta) y - (b\cos\theta + a\sin\theta)x_c + (a\cos\theta - b\sin\theta)y_c = 0\)   ...(12)
   
   The equations (11) and (12) above give the Equations of Rotated and Translated Asymptotic Lines corresponding to Rotated and Translated Hyperbolas.