Questions from previous years IIT Joint Entrance Examination (Indian Institute of Technology)

1. Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

2. The points (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices line on the line y=2x+c. Find c and the remaining vertices.

3. The coordinates A, B, C are (6,3), (-3,5), and (4,-2) respectively and P is any point (x,y). Show that the ratio of the areas of ΔPBC and  ΔABC is

4. Find the condition that the lines ax+by+c=0, bx+cy+a=0, and cx+ay+b=0 are concurrent.

5. A rectangle PQRS has its side PQ parallel to the line y=mx and vertices P, Q, and S  on the lines y=a, x=b, and x=-b respectively. Find the locus of the vertex R.

6. Show that the locus of a point that divides a chord of slope 2 of the parabola y²=4x internally in the ratio 1:2 is a parabola. Find the vertex of this parabola.

7. Through the vertex O of a parabola y²=4x, chords OP and OQ are drawn at right angles to one another. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of PQ.

8. A variable straight line of slope 4 intersects the hyperbola xy=1 at two points. Find the locus of the point which divides the line segment between these two points in the ratio 1:2.