CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.
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1 UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose distances from the points (6, ) and (,-) are in the ratio :. [Ans:(0, ), (0, 5 )] Ans: Point on y-axis (0, y) A(6, ) B(, -) ratio : 6 ( ( y ) y ) 5 On solving we get (0, ) & (0, ). Determine the ratio in which the line x + y - 0 divide the line segment joining the points A (,-) and B (, ).Also find the coordinates of the point of division. [Ans: :, (,- )] Ans : Let the ratio be : Let the co-ordinates of point of division be (x, y) x y () ( ).. (x, y) lies on the line x + y (+) + (-) (+) Ratio is : File downloaded from Page 55
2 y x x 6 (x, y) (, ) 8 8 x x. Find the third vertex of a triangle if its two vertices are (-, ) and (5, ) and mid point of one side is (0, ). (Ans: (-5, ) or (, )) Ans : Let the third vertex be (x, y) If (0,) is mid point of BC then x y 5 0 (or) x - 5 y. (-5, ) If (0,) is mid point of AC then x 0 x (-5. ) or (, ) are possible answers. y y + 6 y (, ). If the vertices of a triangle are (, ), (, -), (-, ) and its area is 5 sq units, find the value(s) of.. [Ans: -, ] Ans: A(, ) B(, -) C(-, ) Area of ABC [x (y -y )+x (y -y ) + x (y -y )] [(--)+(-)+(-)(+)] , File downloaded from Page 56
3 5. The centre of a circle is (x, x + ).Find x if the circle passes through (-,-) and the length of the diameter is 0 units. [Ans: x, - 6 ] Ans : D 0 R 0 (x + ) + (x + + ) 0 (x + ) + (x + ) 00 x + 8x + + x + x + 00 x + 0x x + 0x - 0 x + 6x 6x 0 (x + 6) (x + 6) 0 x, 6 6. If A & B are (-,-) and (,-) respectively, find the co ordinates of P such that 0 AP AB and P lies on the line segment AB. [Ans: (-, - )] Ans : AP AB AP AB AP (i.e) PB AB AP + PB AP : PB : Let P(x, y) x y () ( ) ( ( ) (x, y) (, 6 8 ) 0 ) 8 0. Show that the points (, 0), (, 5), (-, ) and (-, -) taen in order are the vertices of a rhombus. Also find the area of the rhombus. (Ans: sq units) Ans : Let AC be d & BD be d Area dd d 0 d File downloaded from Page 5
4 Area d d x x 6 x sq units. 8. If A, B and P are the points (-, ), (0, -) and (, ) respectively and P is equidistant from A and B, show that Ans : AP PB AP PB ( + ) + ( - ) +( + ) If the points (5, ) and (x, y) are equidistant from the point (, 5), prove that x + y 8x 0y + 0. Ans : AP PB AP PB (5 ) + ( 5) (x ) + (y 5) + x 8x y 0y + 5 x + y 8x - 0y + 0 x + y 8x - 0y If two vertices of an equilateral triangle are (0, 0) and (, 0), find the third vertex.. [Ans:, or, - ] Ans: OA OB AB OA OB AB OA (-0) + 0 OB x + y AB (x-) + y x + y 6x + OA OB AB OA OB & OB AB x + y y - x x + y 6x + O (0, 0) A (, 0) x + -x 6x + 6x x y B(x, y) y File downloaded from Page 58
5 Third vertex is, or,. Find the centre of a circle passing through the points (6, -6), (, -) and (, ).Also find the radius. (Ans: (, -), 5 units) Ans: OAOB OC radius of the circle where O is the centre of the circle and let O be (x, y) OA OB OC OA (x-6) + (y+6) x + y x y + 6 OB (x-) + (y+) x + y 6x + + y + OC (x-) + (y-) x + y 6x + - 6y + OA OB x + y x + y + x + y 6x + y + 58 x + y + 6x y x - y + 0 x - y () x + y 6x + + y + x + y 6x + 6y + 6x + y x 6y + 8 y + 6y y 0 y -...() Substituting we get - x x - x (x, y) (, -) Diameter + 6() (-) Radius 5 5 units. The two opposite vertices of a square are (-, ) and (, ). Find the coordinates of the other two vertices. (Ans: (, 0), (, )) Ans : AB BC AD BC (x + ) + (y-) (x-) + (y-) x + x + + y y + x 6x + + y y + x y + 5-6x y + 8x 5 8x 8 x On substituting in (x-) + (y-) + (x+) + (y-) (- -) + ( - ) We get y or 0. B (, ) or (, 0) A(-,) B(x, y) D (x,y ) C(,) File downloaded from Page 5
6 AD DC AD DC (x + ) + (y -) (x -) + (y -) x. On substituting in (x + ) + (y -) + (x -) + (y -) 6 We get y 0 or. D (, ) or (, 0) the opposite vertices are (, ) & (, 0). Find the coordinates of the point P which is three fourth of the way from A (, ) to B (-, 5). (Ans: (-, ) Ans : Hint: Ratio AP:PB :. The midpoint of the line joining (a, ) and (-, b) is (, a +).Find the values of a & b. (Ans: a, b ) Ans : A(a, ) P(, a + ) B(-, b) a We get a & b. b & a 5. Find the distance between the points (b + c, c + a) and (c + a, a + b). (Ans : Ans : Use distance formula a b c ab bc ) 6. Find the relation between x and y when the point (x,y) lies on the straight line joining the points (,-) and (,) [ Hint: Use area of triangle is 0] Ans : Hint: If the points are on straight line, area of the triangle is zero.. Find the distance between (cos, sin ) and (sin, -cos ). (Ans: ) Ans : cos Sin Sin cos On simplifying we get 8. Find the distance between (a cos5 o, 0) (0, a cos65 o ). (Ans: a ) Ans : Proceed as in sum no.. File downloaded from Page 60
7 . The vertices of a ABC are A(, 6), B(. 5) and C(, ). A line is drawn to intersect sides AB and AC at D and E respectively, such that. AD AB AE AC Calculate the area of 5 the ADE and compare it with the area of ABC. (Ans: sq units; :6) Ans : Hint : AD AB AE AC AD : DB : & AE : EC : Find D & E and find area of triangle ADE and triangle ABC and compare. 0. Plot the points A(,0) and B (6,0) on a graph paper. Complete an equilateral triangle ABC such that the ordinate of C be a positive real number.find the coordinates of C (Ans: (, ) Ans : Proceed by taing C(x, y) AC BC AB. Find the ratio in which the line segment joining A(6,5) and B(,-) is divided by the line y (Ans::5) Ans : Let the ratio be : x y On solving we get : 5. The base BC of an equilateral triangle ABC lies on the y-axis. The coordinates of C are (0,-). If the origin is the midpoint of BC find the coordinates of points A and B. Ans : Hint : The point A will lie on the x axis. Find A using AB BC AC. Coordinates of B (0, ) File downloaded from Page 6
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