#### Level – I

1. Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

2. P is any point on the auxiliary circle of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ and Q is its corresponding point on the ellipse. Find the locus of the point which divides PQ in the ratio of 1 : 2.

3. Find the locus of point of intersection of the tangents at the end point of chord of ellipse $ \displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ , which subtends an angle α at origin.

4. Consider the family of circles x^{2} + y^{2} = r^{2} , 2 < r < 5. if on the first quadrant, the common tangent to a circle of this family and the ellipse 4x^{2} + 25y^{2} = 100 meets the coordinate axes at A and B, then find the equation of the locus of the mid point of AB.

5. Find the latus rectum, the eccentricity, and the coordinates of the foci, of the ellipse 9x^{2} + 5y^{2}– 30y = 0.

6. Find the lengths and equations of the major and minor axes of the ellipse 5x^{2}+5y^{2} + 6xy – 8 = 0.

7. P, Q are two point on ellipse x^{2} + 4y^{2} = 4 such that PQ touches a fixed circle x^{2} + y^{2} – 2x = 0 if α and β be the eccentric angle of P and Q. Prove that secα + secβ = 2.

8. Find the locus of the intersection of the tangents to the ellipse $ \displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ if the difference of the eccentric angles of their points of contact is 2α .

9. The tangent at a point P of the ellipse $ \displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ meets the major axis at B and the ordinate from it meets the major axis at A. If Q is a point on the line AP such that AQ = AB, find the locus of Q.

10. From any point M two normals are drawn to the ellipse $ \displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ , where a > b, which are at right angles and their corresponding tangents meet at N. Find the locus of the mid-point of MN.