# ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.

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1 Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line joining the mid-points of the diagonals of a quadrilateral are concurrent. ( 3 ) If (, 3 ), ( 4, 5 ) and ( a, ) are the vertices of a right triangle, find a. [ Ans: 3, 7 ] ( 4 ) Find the circumcentre of the triangle with vertices ( - 1, 1 ), ( 0, - 4 ) and ( - 1, - 5 ) and deduce that the circumcentre of the triangle whose vertices are (, 3 ), ( 3, - ) and (, - 3 ) is the origin. [ Ans: ( - 3, - ) ] ( 5 ) For which value of a would the area of a triangle with vertices ( 5, a ), (, 5 ) and (, 3 ) be 3 units? [ Ans: For any a R ] ( 6 ) Find the area of the triangle whose vertices are ( l, l ), ( m, m ) and (n, n ) if l m n. [ Ans: l ( l - m ) ( m - n ) ( n - l ) l ] ( 7 ) Find the area of the triangle whose vertices are ( 5, 3 ), ( 4, 5 ) and ( 3, 1 ) and show that the triangle whose vertices are ( -, ), ( - 3, 4 ) and ( - 4, 0 ) has the same area. ( 8 ) Find the area of the triangle with vertices ( 5, 3 ), ( 4, 5 ) and ( 3, 1 ) by shifting the origin at ( 5, 3 ). ( 9 ) Prove that the mid-point of the segment joining the two points dividing AB from A in the ratios m : n and n : m is the mid-point of AB.

2 Problems 01 - POINT Page ( 10 ) If P ( 1, ) and Q ( 5, 6 ) divide AB from A in the ratios : 1 and - : 1, find the co-ordinates of A and B. [ Ans: A ( - 1, 0 ), B (, 3 ) ] ( 11) If ( 3, ), ( 4, 5 ) and (, 3 ) are three of the four vertices of a parallelogram, find the co-ordinates of the fourth vertex. [ Ans: ( 5, 4 ), ( 3, 6 ), ( 1, 0 ) ] ( 1 ) Show that the points (, 3 ), ( 4, 5 ) and ( 3, ) can be the vertices of a rectangle and find the co-ordinates of the fourth vertex. [ Ans: ( 5, 4 ) ] ( 13 ) If the mid-points of the sides of a triangle are ( 4, 3 ), ( 5, - 1 ) and (, 7 ), find the vertices of the triangle. [ Ans: ( 7, - 5 ), ( 1, 11 ), ( 3, 3 ) ] ( 14) Find co-ordinates of the centroid, circumcentre and in-centre of the triangle whose vertices are ( 3, 4 ), ( 0, 4 ) and ( 3, 0 ). 8 Ans :, 3, 3,, (, 3 ) ( 15 ) A ( 3, 4 ), B ( 0, - 5 ) and C ( 3, - 1 ) are the vertices of triangle ABC. Determine the length of the altitude from A on BC. ( 16 ) Points B ( 4, 1 ) and C (, 5 ) are given. Find the equation of sets of all points P in the plane such that m BPC = π. Find the set of all such points P. [ Ans: { ( x, y ) l x + y - 6x + 6y + 13 = 0 } - { B, C } ]

3 Problems 01 - POINT Page 3 ( 17 ) If A ( 0, 1 ) and B (, 9 ) are given, find C on 5 Ans : -, -, , 3 3 AB such that AB = 3 AC. ( 18 ) Points A ( x 1, x 1 tan θ 1 ), B ( x, x tan θ ) and C ( x 3, x 3 tan θ 3 ) are given. If the circumcentre of triangle ABC is origin and its cenroid is ( x, y ), prove that x y = cos θ1 + cos θ sin θ1 + sin θ + cos θ3 + sin θ3 ( 0 < θ 1, θ, θ 3 < π and x 1, x, x 3 > 0 ) ( 19 ) Prove that the mid-points of the sides of any quadrilateral are the vertices of a parallelogram. ( 0 ) If D and G are respectively the mid-point of side BC and the centroid in triangle ABC, then prove that ( i ) AB + AC = ( AD + B D ) and ( ii ) AB + BC + CA = 3 ( GA + GB + GC ) ( 1 ) Prove that the coordinates of all three vertices of an equilateral triangle cannot be rational numbers. ( ) If A, B, C and P are distinct non-collinear points of the plane, prove that Area of PAB + Area of PBC + Area of PCA Area of ABC ( 3 ) If P is a point on the segment joining A ( 3, 5 ) and B ( - 5, 1 ) such that the area of POQ is 6 units where O is the origin and Q is the point ( -, 4 ), then find the coordinates of P. Ans : ( 1, 4 ), 19 -, ( 4 ) Prove that the area of the triangle formed by the mid-points of the sides of the triangle is one-fourth that of the original triangle.

4 Problems 01 - POINT Page 4 ( 5 ) P ( - 5, 1 ), Q ( 3, 5 ), B ( 1, 5 ) and C ( 7, - ) are four points in the plane. The point A divides the segment PQ in the ratio λ : 1 from P. If the area of triangle ABC is units, find the value of λ and also the co-ordinates of A. Ans : λ = 7, Α, 9 ; λ =, Α, ( 6 ) If the co-ordinates of A, B and P are ( x 1, y 1 ), ( x, y ) and ( x, y ) respectively, and if A - P - B, then prove that x + y lies between x 1 + y 1 and x + y. ( 7 ) Chord CD is parallel to the diameter AB of a given circle. P is any point on AB. Prove that PA + PB = PC + PD. ( 8 ) If P is a variable point on the circumcircle of an equilateral triangle ABC, prove that the value of AP + BP + CP is independent of the position of the point P. ( 9 ) A straight line l intersects the lines BC, CA and AB along the sides of BP CQ AR ABC respectively at P, Q and R. Prove that = 1. PC QA RB triangle ( 30 ) A variable rod of length l has one end A on X-axis and another end B on Y-axis. Prove that the equation of the set of points P which divide AB in the ratio 1 : from A is 9x + 36y = 4l. ( 31 ) If A is ( a cos α, a sin α ) and B is ( a cos β, a sin β ), then find AB. Ans : α - β a sin ( 3 ) A ( a, b ) and B ( c, d ) are two points. If AB subtends an angle of measure θ at the ac + bd origin, then prove that cos θ =. ( a + b ) ( c + d )

5 Problems 01 - POINT Page 5 ( 33 ) If P ( at, at ), Q independent of t. a -, t a t and S ( a, 0 ) are three points, show that is SP SQ ( 34 ) For which value of k would the points ( k, - k ), ( - k + 1, k ) and ( k, 6 - k ) be distinct and collinear? [ Ans: - 1 ] ( 35 ) A is ( - 4, 0 ) and B ( 4, 0 ). Find the locus of a point P such that the difference of its distances from A and B is 4. [ Ans: 3x - y = 1 ] ( 36 ) If the distance between the centroid and incentre of the triangle with vertices ( - 36, 7 ), 5 ( 0, 7 ) and ( 0, - 8 ) is 05 k, then find the value of k. 3 Ans : 1 k = 5 ( 37 ) Prove that the locus of in-centre of a variable triangle whose one vertex is the origin and the other two vertices are on the co-ordinate axes is the set S = { ( x, y ) l x = y, x, y R } - { ( 0, 0 ) }. ( 38 ) Prove that the locus of circumcentre of a variable triangle whose one vertex is the origin and the other two vertices are on the co-ordinate axes such that the side opposite to the origin is of constant length l is the set S = { ( x, y ) l 4 x + 4y = l l l l l } -, 0, -, 0, 0,, 0, -. ( 39 ) Prove that the locus of centroid of a variable triangle whose one vertex is the origin and the other two vertices are on the co-ordinate axes such that the side opposite to the origin is of constant length l is the set S = { ( x, y ) l 9 x + 9y = l l l l l } -, 0, -, 0, 0,, 0,

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