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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 x2a2 - y2b2 = 1 is - x2a2 + y2b2 = 1

The transverse axes of the hyperbola x2a2 - y2b2 = 1 is along x-axis and its length = 2a.

The conjugate axes of the hyperbola x2a2 - y2b2 = 1 is along y-axis and its length = 2b.

Therefore, the hyperbola conjugate to x2a2 - y2b2 = 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 x2a2 - y2b2 = 1 is - x2a2 + y2b2 = 1

Thus, the hyperbolas x2a2 - y2b2 = 1 and - x2a2 + y2b2 = 1 are conjugate to each other.

The eccentricity of the conjugate hyperbola is given by a2 = b2(e2 - 1).


Now we will come across various results related to the hyperbola x2a2 - y2b2 = 1 ……………. (i) and its conjugate - x2a2 + y2b2 = 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 = a2+b2a2 or, b2 = a2(e2 - 1) and its conjugate hyperbola (ii) is e = b2+a2b2 or, a2 = b2(e2 - 1).

7. The length of the latusrectum of the hyperbola (i) is 2b2a and its conjugate hyperbola (ii) is 2a2b.

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

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