Explanation and method to remember all rules of standard hyperbolas

NCERT EXERCISE 11.4

In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

Question 1. \frac{x^2}{16}-\frac{y^2}{9}=1

Question 2. \frac{y^2}{9}-\frac{x^2}{27}=1

Question 3. 9y^2-4x^2=36

Question 4. 16x^2-9y^2=576

Question 5. 5y^2-9x^2=36

Question 6. 49y^2-16x^2=784

In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions.

Question 7. Vertices (± 2, 0), foci (± 3, 0)

Question 8. Vertices (0, ± 5), foci (0, ± 8)

Question 9. Vertices (0, ± 3), foci (0, ± 5)

Question 10. Foci (± 5, 0), the transverse axis is of length 8.

Question 11. Foci (0, ±13), the conjugate axis is of length 24.

Question 12. Foci (\pm 3 \sqrt{5}, 0) , the latus rectum is of length 8.

Question 13. Foci (± 4, 0), the latus rectum is of length 12

Question 14. vertices (± 7,0), e = \frac{4}{3}

Question 15. Foci (0, \pm \sqrt{10}), passing through (2, 3)