Rectangular Hyperbola

1. Hyperbolas having Both Transverse and Conjugate Axis of Same Length are called Rectangular Hyperbolas.
2. Following give the Equations of Axis Aligned Rectangular Hyperbolas having Centers at Origin and Semi-Transverse / Semi-Conjugate Axis Length \(a\)
   
   \(x^2-y^2=a^2\)   (\(X\)-Transverse Hyperbolas)...(1)
   
   \(y^2-x^2=a^2\)   (\(Y\)-Transverse Hyperbolas)...(2)
   
   
3. Following give the Equations of Axis Aligned Rectangular Hyperbolas having Centers at \((x_c,y_c\)) and Semi-Transverse / Semi-Conjugate Axis Length \(a\)
   
   \({(x-x_c)}^2-{(y-y_c)}^2=a^2\)    \(\Rightarrow x^2 - y^2 -2x_c -2y_c+{x_c}^2+{y_c}^2-a^2=0\)   (\(X\)-Transverse Hyperbolas)...(3)
   
   \({(y-y_c)}^2-{(x-x_c)}^2=a^2\)    \(\Rightarrow y^2 -x^2 -2x_c -2y_c +{x_c}^2+{y_c}^2-a^2=0\)   (\(Y\)-Transverse Hyperbolas)...(4)
   
   
4. Following give the Equations of Rectangular Hyperbolas having Centers at Origin and Semi-Transverse / Semi-Conjugate Axis Length \(a\) Rotated by 45° and 135° from Positive Direction of \(X\)-Axis
   
   \(xy=\frac{a^2}{2}\)   (Hyperbolas Rotated by 45°)...(5)
   
   \(xy=-\frac{a^2}{2}\)   (Hyperbolas Rotated by 135°)...(6)
   
   
5. Following give the Equations of Rectangular Hyperbolas having Centers at \((x_c,y_c\)) and Semi-Transverse / Semi-Conjugate Axis Length \(a\) Rotated by 45° and 135° from Positive Direction of \(X\)-Axis
   
   \((x-x_c)(y-y_c)=\frac{a^2}{2}\)    \(\Rightarrow xy - y_cx - x_cy +x_cy_c - \frac{a^2}{2}=0\)   (Hyperbolas Rotated by 45°)...(7)
   
   \((x-x_c)(y-y_c)=-\frac{a^2}{2}\)    \(\Rightarrow xy - y_cx - x_cy +x_cy_c + \frac{a^2}{2}=0\)   (Hyperbolas Rotated by 135°)...(8)
   
   
6. Rotating any Rectangular Hyperbola by 90° gives its Conjugate Hyperbola. Hence the Hyperbolas represented by equations (1) and (2) are Conjugates of each other. So are Hyperbolas represented by equations (3) and (4), equations (5) and (6) and equations (7) and (8).
7. Following are Some important Properties of Equations and Parameters of Rectangular Hyperbolas
   1. The Absolute Value of Co-efficients of terms \(x^2\) and \(y^2\) are same (0 for Hyperbolas Rotated by 45° or 135°).
   2. Coordinates of Center: \((-x_c,-y_c)\)
   3. Distance Between 2 Foci: \(F=\sqrt{L^2+L^2} \Rightarrow F=\sqrt{2}L \Rightarrow 2c=2\sqrt{2}a\Rightarrow c=\sqrt{2}a\)
   4. Eccentricity: \(e=\frac{F}{L}=\frac{c}{a}=\frac{\sqrt{2}a}{a}=\sqrt{2}\)
   5. Length of Latus Recta: \(L=2a\)
   6. Equation of Asymptotes:
      
      \((x-(-x_c))=(y-(-y_c)),\hspace{.5cm}(x-(-x_c))=-(y-(-y_c))\)   (For Axis Aligned Hyperbolas)
      
      \(x=-x_c,\hspace{.5cm} y=-y_c\)   (For Hyperbolas Rotated By 45° or 135°)