What is conjugate hyperbola?
If the transverse axis and conjugate axis of any hyperbola be respectively the conjugate axis and transverse axis of another hyperbola then the hyperbolas are called the conjugate hyperbola to each other.
The conjugate hyperbola of the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 is  \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
The transverse axes of the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 is along xaxis and its length = 2a.
The conjugate axes of the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 is along yaxis and its length = 2b.
Therefore, the hyperbola conjugate to \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 will have its transverse and conjugate axes along y and xaxes respectively while the length of transverse and conjugate axes will be 2b and 2a respective.
Therefore, the equation of the hyperbola conjugate to \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 is  \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
Thus, the hyperbolas \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 and  \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 are conjugate to each other.
The eccentricity of the conjugate hyperbola is given by a\(^{2}\) = b\(^{2}\)(e\(^{2}\)  1).
Now we will come across various results related to the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 ……………. (i) and its conjugate  \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 ………………. (ii).
1. The coordinates of the centre of both the hyperbola (i) and its conjugate hyperbola (ii) are (0, 0).
2. The coordinates of the vertices of the hyperbola (i) are (a, 0) and (a, 0) and its conjugate hyperbola (ii) are (0, b) and (0, b).
3. The coordinates of the foci of the hyperbola (i) are (ae, 0) and (ae, 0) and its conjugate hyperbola (ii) are (0, be) and (0, be).
4. The length of the transverse axis of the hyperbola (i) is 2a and its conjugate hyperbola (ii) is 2b.
5. The length of the conjugate axis of the hyperbola (i) is 2b and its conjugate hyperbola (ii) is 2a.
6. The eccentricity of the hyperbola (i) is e = \(\sqrt{\frac{a^{2} + b^{2}}{a^{2}}}\) or, b\(^{2}\) = a\(^{2}\)(e\(^{2}\)  1) and its conjugate hyperbola (ii) is e = \(\sqrt{\frac{b^{2} + a^{2}}{b^{2}}}\) or, a\(^{2}\) = b\(^{2}\)(e\(^{2}\)  1).
7. The length of the latusrectum of the hyperbola (i) is \(\frac{2b^{2}}{a}\) and its conjugate hyperbola (ii) is \(\frac{2a^{2}}{b}\).
8. The equation of the transverse axis of the hyperbola (i) is y = 0 and its conjugate hyperbola (ii) is x = 0.
9. The equation of the conjugate axis of the hyperbola (i) is x = 0 and its conjugate hyperbola (ii) is y = 0.
● The Hyperbola
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