Derivation of Implicit Coordinate Equation for Arbitrarily Rotated and Translated Hyperbolas

1. As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Hyperbolas, the Standard Coordinate Equation for Hyperbolas having their Transverse Axes Parallel to the \(X\)-Axis, Center at Origin, Length of Semi-Transverse Axis \(a\) and Length of Semi-Conjugate Axis \(b\) is given as
   
   \(\frac{x^2}{a^2} - \frac{y^2}{b^2} =1 \)   ...(1)
   
   The Equation of an Arbitrary Rotated Hyperbola with its Center Translated to a given Point can be obtained using following steps in order
   1. Rotate the Standard Coordinate Equation of Hyperbola as given in equation (1) by the Arbitrary Angle \(\theta\).
   2. Translate the Equation of Rotated Hyperbola to the given Point.
2. The following derives the Equation of Rotated Hyperbola from Standard Coordinate Equation of Hyperbola given in equation (1)
   
   Multiplying equation (1) by \(a^2b^2\)on Both Sides we get
   
   \(b^2x^2- a^2y^2 =a^2b^2 \)
   
   \(\Rightarrow b^2x^2- a^2y^2 - a^2b^2 = 0 \)   ...(2)
   
   Rotating the equation (2) above Counter Clockwise with angle \(\theta\) with respect to Positive Direction of \(X\) Axis we get
   
   \(b^2{(x \cos\theta + y \sin\theta)}^2- a^2{(y \cos\theta - x \sin\theta)}^2 - a^2b^2 = 0 \)
   
   \(\Rightarrow b^2(x^2 \cos^2\theta + y^2 \sin^2\theta + 2xy\sin\theta\cos\theta )- a^2(y^2\cos^2\theta + x^2\sin^2\theta - 2xy\sin\theta\cos\theta ) - a^2b^2 = 0 \)
   
   \(\Rightarrow b^2x^2\cos^2\theta + b^2y^2 \sin^2\theta + 2b^2xy\sin\theta\cos\theta - a^2y^2\cos^2\theta - a^2x^2\sin^2\theta + 2a^2xy\sin\theta\cos\theta - a^2b^2 = 0 \)
   
   \(\Rightarrow (b^2\cos^2\theta - a^2\sin^2\theta)x^2 + (2b^2\sin\theta\cos\theta + 2a^2\sin\theta\cos\theta)xy + (b^2 \sin^2\theta - a^2\cos^2\theta)y^2 - a^2b^2 = 0 \)
   
   \(\Rightarrow (b^2\cos^2\theta - a^2\sin^2\theta)x^2 + (2\sin\theta\cos\theta(b^2 + a^2))xy + (b^2 \sin^2\theta - a^2\cos^2\theta)y^2 - a^2b^2 = 0\)
   
   \(\Rightarrow (b^2\cos^2\theta - a^2\sin^2\theta)x^2 + ((b^2 + a^2)\sin2\theta)xy + (b^2 \sin^2\theta - a^2\cos^2\theta)y^2 - a^2b^2 = 0\)   ...(3)
   
   The equation (3) above gives the Equation of the Rotated Hyperbola having Center at Origin
   
   Setting \(b^2\cos^2\theta - a^2\sin^2\theta=\mathbf{A}\), \((b^2 + a^2)\sin2\theta=\mathbf{B}\) and \(b^2 \sin^2\theta - a^2\cos^2\theta=\mathbf{C}\) in equation (3) above we get
   
   \(Ax^2 + Bxy + Cy^2 - a^2b^2 = 0\)   ...(4)
   
   Now, Translating the Equation of the Rotated Hyperbola as given in equation (4) above so that it's Center is at \((x_c,y_c)\) we get
   
   \(A{(x-x_c)}^2 + B(x-x_c)(y-y_c) + C{(y-y_c)}^2 - a^2b^2 = 0\)
   
   \(\Rightarrow A(x^2+{x_c}^2 - 2x_cx) + B(xy-x_cy-y_cx+x_cy_c) + C(y^2+{y_c}^2-2y_cy) - a^2b^2 = 0\)
   
   \(\Rightarrow Ax^2+A{x_c}^2 - 2Ax_cx + Bxy-Bx_cy-By_cx+Bx_cy_c + Cy^2+C{y_c}^2-2Cy_cy - a^2b^2 = 0\)
   
   \(\Rightarrow Ax^2+ Bxy + Cy^2 - (2Ax_c+By_c)x -(2Cy_c+Bx_c)y +A{x_c}^2+C{y_c}^2+Bx_cy_c -a^2b^2=0\)   ...(5)
   
   Setting \(- (2Ax_c+By_c)=\mathbf{D}\), \(-(2Cy_c+Bx_c)=\mathbf{E}\) and \(A{x_c}^2+C{y_c}^2+Bx_cy_c -a^2b^2=\mathbf{F}\) in equation (5) above we get
   
   \(Ax^2 + Bxy + Cy^2 +Dx +Ey +F=0\)   ...(6)
   
   The equation (5) and (6) above give the Implicit Coordinate Equation of the Hyperbola Rotated Counter Clockwise by Angle \(\theta\) with respect to Positive Direction of \(X\) Axis and having Center at \((x_c,y_c)\) where
   
   \(A = b^2\cos^2\theta - a^2\sin^2\theta\)
   
   \(B = (b^2 + a^2)\sin2\theta\)
   
   \(C = b^2 \sin^2\theta - a^2\cos^2\theta\)
   
   \(D = -(2Ax_c+By_c) = -(2(b^2\cos^2\theta - a^2\sin^2\theta) x_c + ((b^2 + a^2)\sin2\theta) y_c)\)
   
   \(E = -(2Cy_c+Bx_c) = -(2(b^2 \sin^2\theta - a^2\cos^2\theta) y_c + ((b^2 + a^2)\sin2\theta) x_c)\)
   
   \(F = A{x_c}^2+C{y_c}^2+Bx_cy_c -a^2b^2 = (b^2\cos^2\theta - a^2\sin^2\theta) {x_c}^2 + (b^2 \sin^2\theta - a^2\cos^2\theta) {y_c}^2 + ((b^2 + a^2)\sin2\theta) x_cy_c - a^2b^2\)