Practice Problems : Parabola

Level – I

1. Prove that 9x2 – 24xy + 16y2 – 20x – 15y – 60 = 0 represents a parabola. Also find its focus and directrix.

2. Two equal parabola have the same vertex and their axes are at right angle. Prove that the common tangent touches each other at the end of latus rectum.

3. Find the equation of parabola whose latus rectum is 4 units, axis is the line 3x + 4y – 4 = 0 and the tangent at the vertex is the line 4x – 3y + 7 = 0.

4. Find the equations of the tangents to the parabola y2 + 4 = 4x which are equally inclined to co-ordinate axis and also find tangent at the vertex of the parabola.

5. From a variable point R on the line y = 2x + 3 tangents are drawn to the parabola y2 = 4ax touch it at P and Q point. Find the locus of the centroid of the triangle PQR.

6. Tangents are drawn from points of the parabola y2 = 4ax to the parabola y2 = 4b (x – c). Find the locus of the mid point of chord of contact.

7. Prove that the normal chord to a Parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

8. Find the shortest distance between the parabola, y2 = 4x and circle x2+y2 – 24y + 128 = 0.

9. From a point A common tangents are drawn to the circle x2 + y2 = a2/2 and the parabola y2 = 4ax . Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.

10. Show that the locus of the point of intersection of tangents to y2 = 4ax , which intercept a constant length d on the directrix is (y2 – 4ax) (x + a)2 = d2 x2

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