Conjugate Hyperbola

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 x-axis and its length = 2a.

The conjugate axes of the hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 is along y-axis 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 x-axes 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 co-ordinates of the centre of both the hyperbola (i) and its conjugate hyperbola (ii) are (0, 0).

2. The co-ordinates 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 co-ordinates 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






11 and 12 Grade Math

From Conjugate Hyperbola to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.