Practice Problems : Hyperbola

Level – I

1. Find the equation of hyperbola whose eccentricity is 5/4, whose focus is (3, 0) and whose directrix is 4x – 3y = 3.

2. The transverse axes of a rectangular hyperbola is 2c and the asymptotes are the axes of coordinates;
show that the equation of the chord which is bisected at the point (2c, 3c) is 3x + 2y = 12c.

3. Find the equation to the chord of hyperbola 25x² – 16y² = 400
which is bisected at the point (5, 3).

4. If two points P and Q on the hyperbola $ \displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ ,
whose centre C be such that CP is perpendicular to CQ , a < b, then prove that

$ \displaystyle \frac{1}{CP^2}+\frac{1}{CQ^2} = \frac{1}{a^2} -\frac{1}{b^2}$

5. The asymptotes of a hyperbola having centre at the point (1 , 2) are parallel to the lines 2x + 3y = 0 and 3x + 2y = 0. If the hyperbola passes through the point (5, 3)
show that its equation is (2x + 3y – 8)(3x + 2y + 7) = 154.

6. Find the eccentricity of the hyperbola whose latus rectum is half of its transverse axis.

7. Find the equation of hyperbola whose foci are (6, 4) and (-4, 4) and eccentricity is 2.

8. Show that there cannot be any common tangent to the hyperbola $ \displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$ and its conjugate hyperbola.

9. If the normals at four points P ( x_i , y_i ), i = 1 , 2 , 3 , 4 on the rectangular hyperbola xy = c² , meet at the point Q(h , k) , prove that

(i) x₁ + x₂ + x₃ + x₄ = h.

(ii) y₁ + y ₂ + y₃ + y₄ = k.

(iii) x₁ x₂ x₃ x₄ = y₁ y₂ y₃ y₄ = c⁴

10. Let ‘ p ‘ be the perpendicular distance from the centre of the hyperbola
$ \displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$
to the tangent drawn at a point R on the hyperbola. If S and S’ are the two foci of the hyperbola.
Then show that ( RS + RS’) = 4a² (1 + b²/p²).