#### 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 25x^{2} – 16y^{2} = 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 = c^{2} , meet at the point Q(h , k) , prove that

(i) x_{1} + x_{2} + x_{3} + x_{4} = h.

(ii) y_{1} + y _{2} + y_{3} + y_{4} = k.

(iii) x_{1} x_{2} x_{3} x_{4} = y_{1} y_{2} y_{3} y_{4} = c^{4}

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’) = 4a^{2} (1 + b^{2}/p^{2}).