Coordinate Systems, Locus and Straight Line

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Gordon Hart
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1 Coordinate Systems Locus Straight Line. A line makes zero intercepts on - ax - ax it perpendicular to the line. Then the equation (Karnataka CET 00). If p the length if the perpendicular from the origin on the line whose intercepts on the axes are a b then 004) (Karnataka CET 6. If the area of the triangle with vertices ( ) ( ) ( ) in 4 sq. units then a value of (Karnataka CET 004 COMED 006) 7. If (0 -) (0 ) are two vertices of a square the other two vertices are (0 ) (0 -) ( -) (00) ( ) (- ) ( ) ( ) (Karnataka CET 005) (Karnataka CET 00). If the dtance if any point P from the points A B are equal then the locus of P (Karnataka CET 00) 4. The equation of the locus of a point which moves such that 4 times its dtance from the - ax the square of its dtance from the origin 004) (Karnataka CET 8. The equation of the line becting the join of ( -4) (5 ) having its intercepts on the - ax - ax in the ratio : (Karnataka CET 005) 9. If the lines are concurrent then b = (Karnataka CET 006) 0. The coordinate of the foot of the perpendicular drawn from the point ( 4) on the line 9 7 ( 5) ( -5) ( ) (Karnataka CET 007) 5. Equation of the line making equal intercepts on the axes passing through the point ( 4)

2 . The coordinates of the circumcenter of the triangle with vertices ( ) (4 - ) (4 ) are ( ) ( ) ( ) ( ) (COMED 006). The orthocenter of the triangle formed by A ( ) B (- C ( 5) ( 5) (- ) (0 ) ( ) (COMED 007) neither right angled nor osceles 00) 7. If are in G.P with the same common ratio then the points ( ) ( ) ( ) lie on line lie on the ellipse lie on circle are vertices of a triangle 00). If the medians AD BE of the triangle with vertices A(0 b) B(0 0) C(a 0) are mutually perpendicular then b= a= b= 4. The lines form a triangle are only concurrent b = a (COMED 007) are concurrent with one line becting the angle between the other two none of these 00) 5. A line through the point ( ) intersects the lines at the points A B. The equation of the line AB so that the triangle OAB equilateral 8. A square of side a lies above the - ax has one vertex at the origin. The side passing through the origin makes are angle α 0 with the 4 positive direction of ax. The equation of its diagonal not passing through the origin 00) 9. If the equation of the locus of a point equidtant from the points ( ) ( ) then the value of c a b a b a b a b none of these 00) 6. A triangle with vertices (4 0) (- ) ( 5) osceles right angled osceles but not right angled right angled but not osceles a a b b a a b b 00)

3 0. Locus of centroid of the triangle whose vertices are (a cos t a sin t) (b sint t- b cos t) ( 0) where t a parameter 005) 00) 4. If non zero numbers a b c are in HP then the line always passes through a fixed point. (- -) (- ) (C) (-- ) ( - ) 005). The equation of the line passing through the point (4 ) making intercepts on the coordinate axes whose sum - 004). A( -) B(- ) are the vertices of a triangle ABC. If the centroid of th triangle moves on the line then the locus of the vertex C the line 004). If a vertex of a triangle ( ) the midpoints of two sides through th vertex are (- ) ( ) then the centroid of the triangle A line through the point A( 4) such that it intercepts between the axes bected at A. Its equation 006) 6. If (a ) falls inside the angle made by the line >0 >0 then a belongs to ( ) 006) 0 7. Let PS be the median of the triangle with vertices P( ) Q(6 -) R(7 ). The equation of the line passing through ( -) parallel to PS (IIT SC 000) 8. Area of the parallelogram formed by the lines

4 (IIT SC 00) 9. The number of integer values of m for which the coordinates of the point of intersection of the lines also integer 0 4 (IIT SC 00). Orthocentre of the triangle whose vertices are given by the coordinates (0 0) ( 4) (4 0) (5 - ) 4 (IIT SC 00) 0. The incentre of the triangle with the vertices ( ) (0 0) ( 0) 00) (IIT SC 00 AIEEE 4. A triangle formed by the points O(0 0) A(0 ) B( 0). The number of points having integral coordinates (both ) are strictly inside the triangle (IIT SC 00) 5. If the lines are concurrent then k = (Kerala CET 00). A straight line through the origin O meets the parallel lines at points P Q respectively. Then the point O divides the segment PQ in the ratio : : 4 : 4 : (IIT SC 00) 6. The centroid of a triangle formed by the points (0 0) (cos θ sin θ) (sin θ -cos θ) lies on the line. Then θ =. Let P(- 0) Q(0 0) R( ) be three points. Then the equation of the bector of the angle PQR x y 0 x y 0 tan (Kerala CET 00) x y 0 x y 0 (IIT SC 00) 7. The Orthocentre of the triangle formed by (8 0) (4 6) with the origin

5 4) ( -4) (4 ) ( (Kerala CET 00) 4. The centroid of the triangle ( 7) two of its vertices are (4 8) (- 6) third vertex (0 0) (4 7) (7 4) (7 7) (4 4) 8. The foot of the perpendicular form (- ) to the line (- ) ( ) ( ) ( ) (- -) (Kerala CET 00) 9. The locus of the midpoint of the portion of the line which intercepted between the axes (f) none of these (Kerala CET 00) 40. The value of for which the lines meet at a point 0 4. Three vertices of a parallelogram taken in order are (- -6) ( -5) (7 ). The fourth vertex ( 4) ( ) (4 4) (4 ) (0 0) 4. The angle between the lines 44. The inclination of the line passing through the point (- 6) the midpoint pf the line joining the points (4-5) (- 9) 45. A point moves such that the area of the triangle formed by it with the points ( 5) ( -7) sq. units. Then the locus of the point none of these 46. Dtance between the parallel lines (Orsa J.E.E. 004) 47. Orthocentre of the triangle formed by the lines (0 0) (0 ) ( 0) (- ) (Orsa J.E.E. 004) 48. The area of the triangle with vertices at (-4 ) ( ) (4 -) these 004) (Andhra none of Pradesh

6 49. The area of the triangle with vertices at the points (a b+c) (b c+a) (c a+b) 0 a + b + c ab + bc +ca none of these (CEE Andhra 996 J. M.I.E.E 000) 50. If the points ( ) ( 4) were to be on the same side of the line 7 < a < a = 7 a = a < 7 or a > (EAMCET 000) 5. The vertices of a triangle are (6 0) (0 6) (6 6). The dtance between its circumstances centroid (EAMCET 000) 5. If the point divides the join of ( ) ( ) internally then < 0 0 < < > = (EAMCET 000) 5. The coordinates of the image of the origin O w.r.t the line are (- -) ( ) (- -) (EAMCET 000) 54. A straight rod of length 9 units slides with its ends A B always on X Y ax respectively. Then the locus of the centroid of the triangle AOB (EAMCET 000) 55. The area of the triangle formed by the axes the lines in square units 4 (EAMCET 000) 56. The incentre of the triangle formed by the line (EAMCET 00) 57. The lines cut the - ax at A B respectively. A line l drawn through the point ( ) meets the - ax at C in such a way that abscsae of A B Care in A.P. Then the equation of the line l (EAMCET 00) 58. For all value of a b the line passes through the point (- ) ( -) (- ) ( -) (EAMCET 00) 59. If a line perpendicular to forms a triangle with the coordinate axes whose area sq. units then the equation of the line(s) (EAMCET 00) 60. If (- 6) the image of the point (4 ) with respect to the line L = 0 then L =

7 (EAMCET 00) 6. If the lines are concurrent then k = (EAMCET 00) 6. The point P equidtant from A( ) B(- 5) C(5 -). Then PA = (EAMCET 00) 6. Suppose A B are two points on P( ) such that PA = PB then the midpoint of AB (EAMCET 004) 64. The dtance between the points (a cos θ a sin θ) (a cos a sin ) a then θ = (EAMCET 004) 65. If a point P moves such that its dtance from the point A( ) the line are equal then the locus a straight line a pair of straight line a parabola an ellipse (EAMCET 005) 66. The area of the triangle formed by the lines (in square units) 4 6 (EAMCET 005) 67. If PM the perpendicular from P( ) onto the line then the coordinates of M are ( ) (- 4) (C) ( ) (4 - ) (EAMCET 005) 68. The equation of the line perpendicular to passing through the point of intersection of the lines (EAMCET 005) 69. The lines meet in the common point ( ) ( ) ( ) ( ) (EAMCET 006) 70. The consecutive sides of a parallelogram are. One diagonal of the parallelogram. If the other diagonal then a = - b = - c = a = b = - c = 0 a = - b = - c = 0 a = b = c = a = - b = - c = (Kerala CET 005) 7. If (-4 5) one vertex one diagonal of a square then the equation of second diagonal (Kerala CET 005)

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.