IITPrep

Code : MCOR1

MATHEMATICS

CO-ORDINATE GEOMETRY


  1. Given the equation y=Kx of a straight line L1, passing through the origin of coordinates; and a point P (x0,y0) such that 0 < y0 < Kx0, x0 > 0.Through P is drawn a st. line L2 intersecting L1 at C and the x-axis at B, such that 1/BP + 1/PC is maximum. Determine the equation of L2.
  2. In a circle with center O, OY is the radius ^r to chord AB, intersecting AB at X, and C be the point where YZ intersects AB. Show that DX £ CY.
  3. The line L is tangent to circle S at the point A; B and C are points on L on opposite sides of A and the other tangents from B, C to S intersect at a point P. If B, C vary along L in such a way that the product ½AB½.½AC½ is constant, find the locus of P.
  4. The parabola y2=ax cuts the hyperbola x2-y2=2a2 in P and Q and tangent at P to the hyperbola cuts the parabola again in R. Find the area of the curvilinear triangle PQR.
  5. The ordinates of points P and Q on the parabola y2=12x are in the ratio 1:2. Find the locus of the pt of intersection of the normals to the parabola at P and Q.
  6. A variable line L passing through the pt B(2,5) intersects the lines 2x2-5xy+2y2=0 at P and Q. Find the locus of the point R on L such that BP, BR, BQ are in HP.
  7. Given a circle k of center M and radius r, let AB be a fixed diameter of k and let K be a fixed point of segment AM. Denote by t the line segment to K at A. For any chord (other than AB) passing through K construct P and Q as the points of intersection of lines BC and BD with t. Prove that the product AP.AQ remains a constant as the chord CD varies.
  8. A point P moves such that the sum of the distances for P to the coordinate axes is equal to the distance from P to the point (1, -1). (a) Determine the equation of locus of P. (b) Sketch the graph of the equation.
  9. The tangent to the hyperbola y=3/x at the point P in the first quadrant meets the x-axis at A and the y-axis at B. Prove that the area of triangle AOB, where O is the origin, is independent of the position of P.
  10. Let P be the point on the parabola y=x2+5 which is nearest to line 2x+y-1=0.Find the distance from P to the line.
  11. The normal at any point P on the hyperbola x2/a2 - y2/b2=1 meet the x-axis and y-axis at L and M. Show that PL/PM is independent of the position of P.
  12. A parabola is drawn to pass through A, B the ends of a diameter of x2+y2=16 and to have as directrix a tangent to its director circle, then Prove that the locus of focus of parabola is an ellipse.
  13. Let ABCD be a square in which A lies on the posive y-axis and B lies on the posive x-axis. If D is the point (12,17) find the coordinates of C.
  14. Given a point (a, b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a, b), another on the x-axis and the third on the line y = x.