COORDINATE GEOMETRY. 13. If a line makes intercepts a and b on the coordinate axes, then its equation is x y = 1. a b


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1 NOTES COORDINATE GEOMETRY 1. If A(x 1, y 1 ) and B(x, y ) are two points then AB = (x x ) (y y ) 1 1 mx nx1 my ny1. If P divides AB in the ratio m : n then P =, m n m n. x1 x y1 y 3. Mix point of AB is,. 4. A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) are the vertices of a triangle ABC then the centroid G = x1 x x3 y1 y y3, The line joining a vertex of a triangle to the mid point of the opposite side is called a median. The medians of a triangle are meeting at a point, called the centroid G of the triangle. y y1 6. The slope of the line joining the points (x 1, y 1 ) and (x, y ) is m =. x x 1 7. If a line makes an angle with the positive Xaxis in the positive direction, then its slope m = tan. a 8. The slope of the line ax + by + c = 0 is m = (if b 0) b 9. Let m 1 and m be the slopes of two nonvertical lines L 1 and L. If L 1 L then m 1 = m and if L 1 L then m 1 m = The equation the line with slope m and intercept on Y axis as c is y = mx + c. 11. The equation of the line passing through (x 1, y 1 ) and having slope m is y y 1 = m(x x 1 ). 1. The equation of the line passing through two points (x 1, y 1 ) and (x, y ) is (x x 1 )(y y 1 ) = (y y 1 )(x x 1 ) 13. If a line makes intercepts a and b on the coordinate axes, then its equation is x y = The area of the triangle formed by the line x y 1 = 1 with the coordinate axes is ab. 15. The general form of the equation of a line is ax + by + c = 0 (a + b 0). 16. Slope of X  axis is 0 ; Slope of a horizontal line is 0 Slope of Y axis is not defined Slope of a vertical line is not defined 17. Equation of X axis is y = 0. Equation of the horizontal line passing through (h, k) is y = k. Equation o f Y axis is x = 0. Equation of the vertical line passing through (h, k) is x = h. 18. To find the point of intersection of two lines, solve the two equations for x and y. If three lines are meeting at one point then the lines are concurrent and the point is called the point of concurrence of the three lines. 19. The equation of a line parallel to ax + by + c = 0 is in the form ax + by + K = The equation of a line perpendicular to ax + by + c = 0 is in the form bx ay + K = 0. Mathiit learning Pvt Ltd 36
2 WORKED EXAMPLES 1. If A(, 1), B(1, ), C(0.4), P(x, y) and PA + PB = PC then find the value of x + 9y Sol: (x ) + (y 1) + (x + 1) + (y + ) = [(x 0) + (y 4) ] x 4x y y x + x y + 4y + 4 = x + y y x 18y = 0 x + 9y = 11. If the point of intersection of the lines x + 3y 1 = 0 and x y + 7 = 0 lies on the line Sol: x + Ky 16 = 0, then find K. x + 3y  1 = 0 x y + 7 = 0 + 5y  8 = 0 y = 8 5 x = 0 x = point of intersection is, K 16 = K 80 = 0 K = Find the point on the line x + y + 7 = 0 equidistant from (0, 1), (, 1) Sol: Let the point be P(x, y) Then x + y + 7 = 0.. (1) (x 0) + (y 1) = (x ) + (y 1) x + y y + 1 = x 4x y y + 1 4x = 4 x = 1 from (1), 1 + y + 7 = 0 and so, y = 8 P = (1, 8). 4. Find the equation of the line passing through the point of intersection of the lines x y = 0 and x + y 5 = 0 and perpendicular to the line 3x y + 4 = 0 Sol: x y = 0.. (1) x + y 5 = 0 3x y + 4 = 0 (1) x gives 4x y = 0 x + y 5 = 0 5x  5 = 0 x = 1 P = (1, ) y = () (3) (From ()) A line perpendicular to (3) is x + 3y + K = K = 0 K = 8 Hence, the required line is x + 3y 8 = 0 Mathiit learning Pvt Ltd 37
3 5. Find the distance of the point (, 1) from the line 3x + y 18 = 0 measured along the line x y 1 = 0. Sol: By solving the two equations, We get B = (4, 3) Let A = (, 1) AB = (4 ) (3 1) 8 3x + y  18 = 0 B X  y  1 = 0 A X (,1) (4,3) 6. D (, 1), E (1, ) and F(3, 3) are the midpoints of the sides BC, CA, AB of a triangle ABC. Then find the vertices of triangle ABC. Sol: Let A = (h, k). Then AFDE is a parallelogram and so, mid point of AD = mid point of EF. h, k 1 3 1, 3 h = 1, k = 3 and A = (1, 3) Similarly, B = (6, 6) and C = (, 4) A (3,3)F E (1,) B D (,1) C 7. Find the equation of the line which passes through (3, 4) and is such that the portion of it intercepted by the coordinate axes is divided by the point in the ratio : 3 Sol: From the figure 3a 0 b 0 P =, (3,4) 3 3 a = 5, b = 10 Equation of the line is x y 1 i.e., x + y = Y B (0,b) O 3 P (3,4) A (a,0) X 8. ABCD is a square. The coordinates of opposite vertices A and C are (1, 3), (3, 7). Then find the equation of BD. Sol: P = mid point of AC =, = (, 5) Slope of AC = Slope of BD is m = 1 Equation of BD is y 5 = 1 (x ) i.e., y 10 = x + or x + y 1 = 0. D A (1,3) P 0 90 C (3,7) B Mathiit learning Pvt Ltd 38
4 LEVEL I 1. Show that the points (e, ), (e, 3 ), (3e, 5 ) are collinear.. The points (3, ), (3, ), (0, h) are the vertices of an equilateral triangle. If h <, find h. 3. A (4, 3), B (6, 1) and C (, 5) are the vertices of a triangle. Find the length of the median through A. 4. Find the condition that the points (7, 5), (3, 6), (x, y) to be collinear. 5. A triangle with vertices (4, 0), (1, 1), (3, 5) is. (AIEEE 00). 6. The point P, if Q = (, 4) is the point on the line segment OP such that OQ = 1 OP is., 3 O being the origin. 7. If the point P(x, y) be equidistant from the points A(a + b, a b) and B(a b, a + b) then show that x=y. 8. If A(1, ), B(, 3), C(, 3), P(x, y) and PA + PB = PC, then find the value of 7x 7y. 9. A square has two opposite vertices at (, 3) and (4, 1) then find the length of the side. 10. If (1, 1), (3, 4) are two adjacent vertices of a parallelogram with center (5, 4) then find the remaining vertices If the mid point of join of (x, y + 1) and (x + 1, y + ) is,, then find the mid point of join of (x 1, y + 1) and (x + 1, y 1). 1. One vertex of a rectangle of sides 3 and 4 lies at the origin. The coordinates of the opposite vertex may be (3, 4). (True/False). 13. Show that the points (1, 1), ( 3, 3 ), (0, 31) are the vertices of a right angled isosceles triangle. 14. The medians AD and BE of ABC with vertices A(o, b), B(0,0), C(a,0) are mutually perpendicular if a= 15. Find the value of K such that the line 3x + 4y + 5 K (x + y + 3) = 0 is parallel to Y axis. 16. If (a, b) is the middle point of the intercept of a line between the coordinate axes, then find the equation of the line. 17. Find the angle subtended by the line joining the points (5, ) and (6, 15) at (0, 0). 18. If the area of the triangle formed by the lines x = 0, y = 0, 3x + 4y = a (a > 0) is 1 then find a. 19. A(1, 1), B(5, 3) are opposite vertices of a square. Find the equation of the other diagonal (not passing through A, B). 0. If the point of intersection of the lines x 3y + 5 = 0 and Kx + 4y + = 0 lies on the line x + 7y 3 = 0, then find K. LEVEL II 1. Prove that the points (3, 1), (, 3), (, ), (1, ) form a square.. Prove that the points (a, b), (0, 0), (a, b), (a, ab) are collinear. 3. One end of a line segment of length 10 is (, 3). If the other end has the abscissa 10 then find its ordinate. 4. Find the equation of the straight line which bisects the intercepts made by the lines x +3y = 6, 3x +y = 6 on the axes Mathiit learning Pvt Ltd 39
5 5. If S = (t, t) and S 1 1 =, t t and A = (1, 0), then find the value of 1 AS AS. 6. Show that the points (, ),(, ),(, ),(, ) where,,, are different real numbers are vertices of a square. 7. If two vertices of an equilateral triangle are (0, 0), (3, 3 ), find the third vertex. 8. Find the intercept on Y axis of the line joining the points (, 3) and (4, ). 9. The line joining the points (p + 1, 1) and (p+1, 3) passes through the point (p, p ) for a value of p = 10. Find the point on the line x 3y = 5 equidistant form (1, ), (3, 4). 11. The triangle formed by the lines x + y = 0, 3x + y 4 = 0, x + 3y 4 = 0 is 1. (4, 5) is a vertex of a square and one of these diagonals is 7x y + 8 = 0. Find the equation of the other diagonal. 13. A line passing through A(1, ) has slope 1. Find the points on the line at a distance of 4 units form A. 14. Find the equation of the line joining the origin to the point of intersection of the lines x y 1and x y If the lines y = 4 3x, ay = x + 10, y + bx + 9 = 0 represent three consecutive sides of a rectangle, then find the value of ab. 16. Find the equation of the line passing through the point of intersection of the lines x y + 5 = 0 and 3x + y + 7 = 0 and perpendicular to the line x y = If a straight line perpendicular to the line x 3y + 7 = 0 forms a triangle with the coordinate axes whose area is 3 sq. units, then find the equation of the line. (EAMCET 00). 18. Find the distance of the point (, 3) from the line x 3y + 9 = 0 measured along the line x y + 1 = If x + 3y + 4 = 0 is the perpendicular bisector of the segment joining the points A(1, ) and B (, ) the find the value of. 0. The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line y = x + c. Then find the remaining vertices. LEVEL III 1. A line AB is divided internally at P(, 4) in the ratio : 3 and at Q (0, 5) in the ratio 3 :. Then find A.. Find the point on the line 3x + y + 4 = 0 which is equidistant from the points (5, 6) and (3, ). 3. A line meets the Xaxis at A and Y axis at B such that centroid of triangle OAB is (1, ). Then find the equation of AB. 4. Given A = (at a a, at), B =, t t and S = (a, 0); prove that SA SB a 5. D(, 1), E(1, ) and F(3, 3) are the midpoints of the sides BC, CA, AB of triangle ABC. Find the equation of the sides of triangle ABC. 6. ABCD is a parallogram. Equations of AB and AD are 4x + 5y = 0, 7x + y = 0 and the equation of the diagonal BD is 11x + 7y 9 = 0. Find the equation of AC. Mathiit learning Pvt Ltd 40
6 7. The equations of the lines cutting off intercepts a, b on the coordinate axes such that a + b = 3, ab = 4 are.. 8. Find the equation of the line which passes through (6, 10) and is such that the portion of it intercepted by the coordinate axes is divided by the point in the ratio 5 :. 9. Find the equation of the line which passes through the point (3, 4) and the sum of its intercepts on the axes is The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x y + 5 = 0 and x + y = 0. If A = (1, ) then find the equation of the side BC. KEY TO LEVEL I x + 4y = 7 5. isosceles and right angled 6. (6, 1) (7, 4), (9, 7) 11. (1, 1) 1. True 14. b x y x + y 8 = KEY TO LEVEL II or x + y 5 = (0, 3 )or (3,  3 ) (4, 1) 11. isosceles 1. x + 7y 31 = (3, 6), (5, ) 14. x + y = x + y + = x + y += KEY TO LEVEL III 0. (, 0) and (4, 4) 1. (6, ). (, ) 3. x + y 6 = x 4y 6 = 0, x y = 0, x y = 0 6. x y = 0 7. x 4y = 4, 4x y + 4 = 0 8. x 3y 4 = 0 9. x + y = 7 or 4x + 3y = x + 3y 40 = 0 WORK SHEET 1. Write the equations of straight lines (i) parallel to X axis and at a distance of 4 and 5 units above and below to Xaxis respectively. (ii) Parallel to Yaxis and at a distance of units, from Yaxis to the right of it. (iii) at a distance of 6 units from Yaxis to the left of it.. The slope of the line joining points 1 1 A ab,ab,b ab,ab is 3. The slope of the line perpendicular to the line joining the points A(a, 0), B(0, b) is.. 4. Find the equation of a line passing through (a, b) and having slope b a. 5. If (, 1), ( 5, 7), ( 5, 5) are the midpoints of the sides of a triangle. Find the equations of the sides of the triangle. Mathiit learning Pvt Ltd 41
7 6. Find the equation of lines having slope 1 and whose (i) y intercept is (ii) x intercept is. 7. Find the equation of a line passing through (5, 4) and parallel to x + 3y + 7 = 0 8. Find the equation of a line passing through (b, a) and perpendicular to x + y = A line intersects the coordinate axis such that the area of the triangle formed is 6 sq.units. Also the sum of the intercepts made by the line is 5, then find the equations of line. 10. Find the equation of the line passing through (1, 1) and making equal intercepts on the coordinate axes. 11. Find the area of the square bounded by the lines x + y =, x y =. 1. Find the equation of the line passing through the intersection of the lines x 5y + 1 = 0, 3x + y = 8 and (i) making equal intercepts on the axes (ii) parallel to x + y = 0 (iii) perpendicular to 4x + 5y + = Find the length of the altitude from A to BC of ABC when A = ( 5, 3), B = ( 3, ), C = (9, 1). 14. If the area of the triangle formed by the lines x =0, y = 0 and 3x + 4y = a is 6 sq.units, then find the value of a. 15. If A (10, 4), B( 4, 9), C(, 1) are the vertices of a triangle, then find the equation so (i) AC, BC, CA (ii) medians through A, B, C (iii) The altitudes through A, B, C (iv) Perpendicular bisectors of the sides AB, BC, CA. KEY TO WORK SHEET 1) (i) y = 4, y =  5 (ii) x =, (iii) x= 6 ) b +b 1 3) 4) bx ay = 0 5) 6x + 7y + 65 = 0, x + 5 = 0, 6x 7y + 79 = 0 6) x+y 4=0, x y =0 7) x + 3y = 0 8) x  y = 0 b a 10) x + y = 0 11) 8 sq.units 1) i) x + y 3 = 0, ii) x + y 3 = 0 iii) 5x 4y 6 =0 13) 14) a = units 9) 3x + y 6 = 0, x + 3y 6= 0 15) (i) 5x + 14y 106 = 0, 5x + y + 11 = 0, 5x 1y = 0 (ii) y 4 = 0, 15x + 16y 84 = 0, 3x y + 4 = 0 (iii) x 5y + 10 = 0, 1x + 5y + 3 = 0, 14x 5y + 3 = 0 (iv) 8x 10y 19 = 0, x 5y + 1 = 0, 4x +10y 111 = 0 Mathiit learning Pvt Ltd 4
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