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Method and examples
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Method
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Find the ratio in which point P divides the line segment AB. Also find the value of x
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1. Distance, Slope of two points
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1. Find the distance between the points `A(5,-8)` and `B(-7,-3)`
2. Find the slope of the line joining points `A(4,-8)` and `B(5,-2)`
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` - `A(5,-8),B(-7,-3)`
- `A(7,-4),B(-5,1)`
- `A(-6,-4),B(9,-12)`
- `A(1,-3),B(4,-6)`
- `A(-5,7),B(-1,3)`
- `A(-8,6),B(2,0)`
- `A(0,0),B(7,4)`
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Find the value of x or y
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3. If distance between the points (5,3) and (x,-1) is 5, then find the value of x.
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- `A(5,3),B(x,-1)`, distance `=5`
- `A(x,-1),B(3,2)`, distance `=5`
- `A(x,2),B(3,-6)`, distance `=10`
- `A(x,1),B(-1,5)`, distance `=5`
- `A(x,7),B(1,15)`, distance `=10`
- `A(1,x),B(-3,5)`, distance `=5`
- `A(x,0),B(4,8)`, distance `=10`
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4. If slope of the line joining points `A(x,0), B(-3,-2)` is `2/7`, find the value of `x`
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Slope =
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- `A(x,0),B(-3,-2)`, slope `=2/7`
- `A(2,x),B(-3,7)`, slope `=1`
- `A(x,5),B(-1,2)`, slope `=3/4`
- `A(2,5),B(x,3)`, slope `=2`
- `A(x,2),B(6,-8)`, slope `=-5/4`
- `A(-2,x),B(5,-7)`, slope `=-1`
- `A(2,3),B(x,6)`, slope `=3/5`
- `A(-3,4),B(5,x)`, slope `=-5/4`
- `A(0,x),B(5,-2)`, slope `=-9/5`
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2. Points are Collinear or Triangle or Quadrilateral form
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Show that the points are the vertices of
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Find `A(0,0), B(2,2), C(0,4), D(-2,2)` are vertices of a square or not
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- `A(1,5),B(2,3),C(-2,-11)` are collinear points
- `A(1,-3),B(2,-5),C(-4,7)` are collinear points
- `A(-1,-1),B(1,5),C(2,8)` are collinear points
- `A(0,-1),B(3,5),C(5,9)` are collinear points
- `A(2,8),B(1,5),C(0,2)` are collinear points
- `A(-1,-1),B(1,5),C(2,8)` are collinear points
- `A(0,-1),B(3,5),C(5,9)` are collinear points
- `A(2,8),B(1,5),C(0,2)` are collinear points
- `A(0,0),B(0,3),C(4,0)` are vertices of a right angle triangle
- `A(-2,-2),B(-1,2),C(3,1)` are vertices of a right angle triangle
- `A(-3,2),B(1,2),C(-3,5)` are vertices of a right angle triangle
- `A(2,5),B(8,5),C(5,10.196152)` are vertices of an equilateral triangle
- `A(2,2),B(-2,4),C(2,6)` are vertices of an isosceles triangle
- `A(0,0),B(2,0),C(-4,0),D(-2,0)` are collinear points
- `A(3,2),B(5,4),C(3,6),D(1,4)` are vertices of a square
- `A(0,0),B(2,2),C(0,4),D(-2,2)` are vertices of a square
- `A(1,-1),B(-2,2),C(4,8),D(7,5)` are vertices of a rectangle
- `A(0,-4),B(6,2),C(3,5),D(-3,-1)` are vertices of a rectangle
- `A(3,0),B(4,5),C(-1,4),D(-2,-1)` are vertices of a rhombus
- `A(2,3),B(7,4),C(8,7),D(3,6)` are vertices of a parallelogram
- `A(1,5),B(1,4),C(-1,3),D(-1,4)` are vertices of a parallelogram
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3. Find Ratio of line joining AB and is divided by P
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1. Find the ratio in which the point P(3/4, 5/12) divides the line segment joining the points A(1/2, 3/2) and B(2, -5)
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- `P(3/4,5/12),A(1/2,3/2),B(2,-5)`
- `P(-1,6),A(3,10),B(6,-8)`
- `P(-2,3),A(-3,5),B(4,-9)`
- `P(3,10),A(5,12),B(2,9)`
- `P(6,17),A(1,-3),B(3,5)`
- `P(12,23),A(2,8),B(6,14)`
- `P(3,10),A(5,12),B(2,9)`
- `P(6,17),A(1,-3),B(3,5)`
- `P(12,23),A(2,8),B(6,14)`
- `P(17/5,47/5),A(5,13),B(1,4)`
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2. Write down the co-ordinates of the point P that divides the line joining A(-4,1) and B(17,10) in the ratio 1:2
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- `A(5,13),B(1,4),m:n=2:3`
- `A(-4,1),B(17,10),m:n=1:2`
- `A(5,12),B(2,9),m:n=2:1`
- `A(2,8),B(6,14),m:n=5:3` Externally
- `A(1,-3),B(3,5),m:n=5:3` Externally
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3. In what ratio does the x-axis divide the join of `A(2,-3)` and `B(5,6)`? Also find the coordinates of the point of intersection.
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divided by
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- `A(2,-3),B(5,6)` divided by x-axis
- `A(1,2),B(2,3)` divided by x-axis
- `A(5,-6),B(-1,-4)` divided by y-axis
- `A(-2,1),B(4,5)` divided by y-axis
- `A(2,1),B(7,6)` divided by x-axis
- `A(2,-4),B(-3,6)` divided by y-axis
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4. Find the ratio in which the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)`? Also find the value of `x`.
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- `P(x,2),A(12,5),B(4,-3)`
- `P(11,y),A(15,5),B(9,20)`
- `P(-3,y),A(-5,-4),B(-2,3)`
- `P(-4,y),A(-6,10),B(3,-8)`
- `P(x,4),A(2,1),B(7,6)`
- `P(x,0),A(2,-4),B(-3,6)`
- `P(0,y),A(2,-4),B(-3,6)`
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4. Find Midpoint or Trisection points or equidistant points on X-Y axis
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1. Find the coordinates of the midpoint of the line segment joining the points `A(-5,4)` and `B(7,-8)`
2. Find the trisectional points of line joining `A(-3,-5)` and `B(-6,-8)`
3. Find the point on the x-axis which is equidistant from `A(5,4)` and `B(-2,3)`
4. Find the point on the y-axis which is equidistant from `A(6,5)` and `B(-4,3)`
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- `A(-5,4),B(7,-8)`
- `A(2,1),B(1,-3)`
- `A(2,1),B(5,3)`
- `A(3,-5),B(1,1)`
- `A(1,-1),B(-5,-3)`
- `A(-7,-3),B(5,3)`
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5. Find Centroid, Circumcenter, Area of a triangle
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1. Find the centroid of a triangle whose vertices are `A(4,-6),B(3,-2),C(5,2)`
2. Find the circumcentre of a triangle whose vertices are `A(-2,-3),B(-1,0),C(7,-6)`
3. Using determinants, find the area of the triangle with vertices are `A(-3,5),B(3,-6),C(7, 2)`
4. Using determinants show that the following points are collinear `A(2,3),B(-1,-2),C(5,8)`
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- `A(4,-6),B(3,-2),C(5,2)`
- `A(3,-5),B(-7,4),C(10,-2)`
- `A(4,-8),B(-9,7),C(8,13)`
- `A(3,-7),B(-8,6),C(5,10)`
- `A(2,4),B(6,4),C(2,0)`
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6. Find the equation of a line using slope, point, X-intercept, Y-intercept
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1. Find the equation of a straight line passing through `A(-4,5)` and having slope `-2/3`
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Slope :
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- `A(-4,5)`,slope`=-2/3`
- `A(4,5)`,slope`=1`
- `A(-2,3)`,slope`=-4`
- `A(-1,2)`,slope`=-5/4`
- `A(0,3)`,slope`=2`
- `A(0,0)`,slope`=1/4`
- `A(5,4)`,slope`=1/2`
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2. Find the equation of a straight line passing through the points `A(7,5)` and `B(-9,5)`
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- `A(7,5),B(-9,5)`
- `A(-1,1),B(2,-4)`
- `A(-5,-6),B(3,10)`
- `A(3,-5),B(4,-8)`
- `A(-1,-4),B(3,0)`
- `A(7,8),B(1,0)`
- `A(6,4),B(-1,5)`
- `A(2,3),B(7,6)`
- `A(-3,4),B(5,-6)`
- `A(0,7),B(5,-2)`
- `A(0,0),B(-4,-6)`
- `A(3,5),B(6,4)`
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4. Find the slope, x-intercept and y-intercept of the line joining the points `A(1,3)` and `B(3,5)`
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- `A(1,3),B(3,5)`
- `A(4,-8),B(5,-2)`
- `A(7,1),B(8,9)`
- `A(4,8),B(5,5)`
- `A(7,8),B(1,0)`
- `A(6,4),B(-1,5)`
- `A(2,3),B(7,6)`
- `A(-3,4),B(5,-6)`
- `A(0,7),B(5,-2)`
- `A(0,0),B(-4,-6)`
- `A(3,5),B(6,4)`
- `A(3,-5),B(-7,9)`
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8. Find the equation of a line passing through point of intersection of two lines and slope or a point
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1. Find the equation of a line passing through the point of intersection of lines `3x+4y=7` and `x-y+2=0` and having slope 5
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Line-1 : ,
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Line-2 : ,
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Slope :
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- Line-1`:x-4y+18=0`,Line-2`:x+y-12=0`,slope`=2`
- Line-1`:2x+3y+4=0`,Line-2`:3x+6y-8=0`,slope`=2`
- Line-1`:x=3y`,Line-2`:3x=2y+7`,slope`=-1/2`
- Line-1`:x-4y+18=0`,Line-2`:x+y-12=0`,slope`=2`
- Line-1`:2x+3y+4=0`,Line-2`:3x+6y-8=0`,slope`=2`
- Line-1`:x=3y`,Line-2`:3x=2y+7`,slope`=-1/2`
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2. Find the equation of a line passing through the point of intersection of lines `4x+5y+7=0` and `3x-2y-12=0` and point `A(3,1)`
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Line-1 : ,
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Line-2 : ,
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- Line-1`:x+y+1=0`,Line-2`:3x+y-5=0`,`A(1,-3)`
- Line-1`:4x+5y+7=0`,Line-2`:3x-2y-12=0`,`A(3,1)`
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9. Find the equation of a line passing through a point and parallel or perpendicular to Line-2
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1. Find the equation of the line passing through the point `A(5,4)` and parallel to the line `2x+3y+7=0`
2. Find the equation of the line passing through the point `A(1,1)` and perpendicular to the line `2x-3y+2=0`
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Line-2 :
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- `A(5,4)`,Line`:2x+3y+7=0`
- `A(1,1)`,Line`:2x-3y+2=0`
- `A(2,3)`,Line`:2x-3y+8=0`
- `A(2,-5)`,Line`:2x-3y-7=0`
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3. Find the equation of the line passing through the point `A(1,3)` and parallel to line passing through the points `B(3,-5)` and `C(-6,1)`
4. Find the equation of the line passing through the point `A(5,5)` and perpendicular to the line passing through the points `B(1,-2)` and `C(-5,2)`
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- `A(1,3),B(3,-5),C(-6,1)`
- `A(4,-5),B(3,7),C(-2,4)`
- `A(-1,3),B(0,2),C(4,5)`
- `A(2,-3),B(1,2),C(-1,5)`
- `A(4,2),B(1,-1),C(3,2)`
- `A(5,5),B(1,-2),C(-5,2)`
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12. Reflection of points about x-axis, y-axis, origin
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Find Reflection of points A(0,0),B(2,2),C(0,4),D(-2,2) and Reflection about X,Y,O
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Reflection about
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- `A(-2,-2),B(-1,2),C(3,1)` and Reflection about x
- `A(2,3),B(7,4),C(8,7),D(3,6)` and Reflection about y
- `A(1,-1),B(-2,2),C(4,8),D(7,5)` and Reflection about o
- `A(3,0),B(4,5),C(-1,4),D(-2,-1)` and Reflection about x,y
- `A(3,2),B(5,4),C(3,6),D(1,4)` and Reflection about y,x
- `A(-1,-1),B(1,5),C(2,8)` and Reflection about y=x
- `A(-3,2),B(1,2),C(-3,5)` and Reflection about y=-x
- `A(0,-1),B(3,5),C(5,9)` and Reflection about x=2
- `A(2,8),B(1,5),C(0,2)` and Reflection about y=2
- `A(0,0),B(2,2),C(0,4),D(-2,2)` and Reflection about x+3y-7=0
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Mode =
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Decimal Place =
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Solution
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Solution provided by AtoZmath.com
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Find the ratio in which point P divides the line segment AB. Also find the value of x calculator
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 1. Find the ratio in which the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)`? Also find the value of x.
2. Find the ratio in which the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)`? Also find the value of y.
3. Find the ratio in which the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)`? Also find the value of y.
4. Find the ratio in which the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)`? Also find the value of y.
5. Find the ratio in which the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)`? Also find the value of x.
6. Find the ratio in which the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of x.
7. Find the ratio in which the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of y.
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Example1. Find the ratio in which the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)`? Also find the value of x.Solution:Method-1 : considering the ratio `m:n`Let the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)` in the ratio `m:n` The given points are `A(12,5),B(4,-3)` `:. x_1=12,y_1=5,x_2=4,y_2=-3` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(x,2)=((m(4)+n(12))/(m+n),(m(-3)+n(5))/(m+n))` `:. (m(-3)+n(5))/(m+n)=2` `:. -3m+5n=2(m+n)` `:. -3m+5n=2m+2n` `:. -3m-2m=2n-5n` `:. -5m=-3n` `:. 5m=3n` `:. m/n=(3)/(5)` :. The point divides the line joining `A(12,5)` and `B(4,-3)` in the ratio `3:5` Putting `m=3,n=5`, we get the coordinates of point P `x=(mx_2+nx_1)/(m+n)` `=(3*4+5*12)/(3+5)` `=(12+60)/(8)` `=(72)/(8)` `=9` Thus, the value of x is `9`
Method-2 : considering the ratio `k:1`Let the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)` in the ratio `k:1` The given points are `A(12,5),B(4,-3)` `:. x_1=12,y_1=5,x_2=4,y_2=-3` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(x,2)=((k(4)+(12))/(k+1),(k(-3)+(5))/(k+1))` `:. (k(-3)+(5))/(k+1)=2` `:. -3k+5=2(k+1)` `:. -3k+5=2k+2` `:. -3k-2k=2-5` `:. -5k=-3` `:. 5k=3` `:. k=(3)/(5)` :. The point divides the line joining `A(12,5)` and `B(4,-3)` in the ratio `3:5` Putting `k=3/5`, we get the coordinates of point P `x=(kx_2+x_1)/(k+1)` `=(3/5*4+12)/(3/5+1)` `=(3*4+12*5)/(3+5)` `=(12+60)/(8)` `=(72)/(8)` `=9` Thus, the value of x is `9`
2. Find the ratio in which the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)`? Also find the value of y.Solution:Method-1 : considering the ratio `m:n`Let the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)` in the ratio `m:n` The given points are `A(15,5),B(9,20)` `:. x_1=15,y_1=5,x_2=9,y_2=20` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(11,y)=((m(9)+n(15))/(m+n),(m(20)+n(5))/(m+n))` `:. (m(9)+n(15))/(m+n)=11` `:. 9m+15n=11(m+n)` `:. 9m+15n=11m+11n` `:. 9m-11m=11n-15n` `:. -2m=-4n` `:. 2m=4n` `:. m/n=(4)/(2)` `:. m/n=(2)/(1)` The point divides the line joining `A(15,5)` and `B(9,20)` in the ratio `2:1` Putting `m=2,n=1`, we get the coordinates of point P `y=(my_2+ny_1)/(m+n)` `=(2*20+1*5)/(2+1)` `=(40+5)/(3)` `=(45)/(3)` `=15` Thus, the value of y is `15`
Method-2 : considering the ratio `k:1`Let the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)` in the ratio `k:1` The given points are `A(15,5),B(9,20)` `:. x_1=15,y_1=5,x_2=9,y_2=20` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(11,y)=((k(9)+(15))/(k+1),(k(20)+(5))/(k+1))` `:. (k(9)+(15))/(k+1)=11` `:. 9k+15=11(k+1)` `:. 9k+15=11k+11` `:. 9k-11k=11-15` `:. -2k=-4` `:. 2k=4` `:. k=(4)/(2)` `:. k=(2)/(1)` The point divides the line joining `A(15,5)` and `B(9,20)` in the ratio `2:1` Putting `k=2`, we get the coordinates of point P `y=(ky_2+y_1)/(k+1)` `=(2*20+5)/(2+1)` `=(40+5)/(3)` `=(45)/(3)` `=15` Thus, the value of y is `15`
3. Find the ratio in which the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)`? Also find the value of y.Solution:Method-1 : considering the ratio `m:n`Let the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)` in the ratio `m:n` The given points are `A(-5,-4),B(-2,3)` `:. x_1=-5,y_1=-4,x_2=-2,y_2=3` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(-3,y)=((m(-2)+n(-5))/(m+n),(m(3)+n(-4))/(m+n))` `:. (m(-2)+n(-5))/(m+n)=-3` `:. -2m-5n=-3(m+n)` `:. -2m-5n=-3m-3n` `:. -2m+3m=-3n+5n` `:. m=2n` `:. m/n=(2)/(1)` The point divides the line joining `A(-5,-4)` and `B(-2,3)` in the ratio `2:1` Putting `m=2,n=1`, we get the coordinates of point P `y=(my_2+ny_1)/(m+n)` `=(2*3+1*-4)/(2+1)` `=(6-4)/(3)` `=(2)/(3)` Thus, the value of y is `2/3`
Method-2 : considering the ratio `k:1`Let the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)` in the ratio `k:1` The given points are `A(-5,-4),B(-2,3)` `:. x_1=-5,y_1=-4,x_2=-2,y_2=3` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(-3,y)=((k(-2)+(-5))/(k+1),(k(3)+(-4))/(k+1))` `:. (k(-2)+(-5))/(k+1)=-3` `:. -2k-5=-3(k+1)` `:. -2k-5=-3k-3` `:. -2k+3k=-3+5` `:. k=2` `:. k=(2)/(1)` The point divides the line joining `A(-5,-4)` and `B(-2,3)` in the ratio `2:1` Putting `k=2`, we get the coordinates of point P `y=(ky_2+y_1)/(k+1)` `=(2*3-4)/(2+1)` `=(6-4)/(3)` `=(2)/(3)` Thus, the value of y is `2/3`
4. Find the ratio in which the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)`? Also find the value of y.Solution:Method-1 : considering the ratio `m:n`Let the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)` in the ratio `m:n` The given points are `A(-6,10),B(3,-8)` `:. x_1=-6,y_1=10,x_2=3,y_2=-8` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(-4,y)=((m(3)+n(-6))/(m+n),(m(-8)+n(10))/(m+n))` `:. (m(3)+n(-6))/(m+n)=-4` `:. 3m-6n=-4(m+n)` `:. 3m-6n=-4m-4n` `:. 3m+4m=-4n+6n` `:. 7m=2n` `:. m/n=(2)/(7)` The point divides the line joining `A(-6,10)` and `B(3,-8)` in the ratio `2:7` Putting `m=2,n=7`, we get the coordinates of point P `y=(my_2+ny_1)/(m+n)` `=(2*-8+7*10)/(2+7)` `=(-16+70)/(9)` `=(54)/(9)` `=6` Thus, the value of y is `6`
Method-2 : considering the ratio `k:1`Let the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)` in the ratio `k:1` The given points are `A(-6,10),B(3,-8)` `:. x_1=-6,y_1=10,x_2=3,y_2=-8` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(-4,y)=((k(3)+(-6))/(k+1),(k(-8)+(10))/(k+1))` `:. (k(3)+(-6))/(k+1)=-4` `:. 3k-6=-4(k+1)` `:. 3k-6=-4k-4` `:. 3k+4k=-4+6` `:. 7k=2` `:. k=(2)/(7)` The point divides the line joining `A(-6,10)` and `B(3,-8)` in the ratio `2:7` Putting `k=2/7`, we get the coordinates of point P `y=(ky_2+y_1)/(k+1)` `=(2/7*-8+10)/(2/7+1)` `=(2*-8+10*7)/(2+7)` `=(-16+70)/(9)` `=(54)/(9)` `=6` Thus, the value of y is `6`
5. Find the ratio in which the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)`? Also find the value of x.Solution:Method-1 : considering the ratio `m:n`Let the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)` in the ratio `m:n` The given points are `A(2,1),B(7,6)` `:. x_1=2,y_1=1,x_2=7,y_2=6` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(x,4)=((m(7)+n(2))/(m+n),(m(6)+n(1))/(m+n))` `:. (m(6)+n(1))/(m+n)=4` `:. 6m+n=4(m+n)` `:. 6m+n=4m+4n` `:. 6m-4m=4n-n` `:. 2m=3n` `:. m/n=(3)/(2)` :. The point divides the line joining `A(2,1)` and `B(7,6)` in the ratio `3:2` Putting `m=3,n=2`, we get the coordinates of point P `x=(mx_2+nx_1)/(m+n)` `=(3*7+2*2)/(3+2)` `=(21+4)/(5)` `=(25)/(5)` `=5` Thus, the value of x is `5`
Method-2 : considering the ratio `k:1`Let the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)` in the ratio `k:1` The given points are `A(2,1),B(7,6)` `:. x_1=2,y_1=1,x_2=7,y_2=6` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(x,4)=((k(7)+(2))/(k+1),(k(6)+(1))/(k+1))` `:. (k(6)+(1))/(k+1)=4` `:. 6k+1=4(k+1)` `:. 6k+1=4k+4` `:. 6k-4k=4-1` `:. 2k=3` `:. k=(3)/(2)` :. The point divides the line joining `A(2,1)` and `B(7,6)` in the ratio `3:2` Putting `k=3/2`, we get the coordinates of point P `x=(kx_2+x_1)/(k+1)` `=(3/2*7+2)/(3/2+1)` `=(3*7+2*2)/(3+2)` `=(21+4)/(5)` `=(25)/(5)` `=5` Thus, the value of x is `5`
6. Find the ratio in which the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of x.Solution:Method-1 : considering the ratio `m:n`Let the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `m:n` The given points are `A(2,-4),B(-3,6)` `:. x_1=2,y_1=-4,x_2=-3,y_2=6` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(x,0)=((m(-3)+n(2))/(m+n),(m(6)+n(-4))/(m+n))` `:. (m(6)+n(-4))/(m+n)=0` `:. 6m-4n=0` `:. 6m=4n` `:. m/n=(4)/(6)` `:. m/n=(2)/(3)` :. The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3` Putting `m=2,n=3`, we get the coordinates of point P `x=(mx_2+nx_1)/(m+n)` `=(2*-3+3*2)/(2+3)` `=(-6+6)/(5)` `=(0)/(5)` Thus, the value of x is `0`
Method-2 : considering the ratio `k:1`Let the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `k:1` The given points are `A(2,-4),B(-3,6)` `:. x_1=2,y_1=-4,x_2=-3,y_2=6` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(x,0)=((k(-3)+(2))/(k+1),(k(6)+(-4))/(k+1))` `:. (k(6)+(-4))/(k+1)=0` `:. 6k-4=0` `:. 6k=4` `:. k=(4)/(6)` `:. k=(2)/(3)` :. The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3` Putting `k=2/3`, we get the coordinates of point P `x=(kx_2+x_1)/(k+1)` `=(2/3*-3+2)/(2/3+1)` `=(2*-3+2*3)/(2+3)` `=(-6+6)/(5)` `=(0)/(5)` Thus, the value of x is `0`
7. Find the ratio in which the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of y.Solution:Method-1 : considering the ratio `m:n`Let the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `m:n` The given points are `A(2,-4),B(-3,6)` `:. x_1=2,y_1=-4,x_2=-3,y_2=6` Using section formula `P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))` `:.P(0,y)=((m(-3)+n(2))/(m+n),(m(6)+n(-4))/(m+n))` `:. (m(-3)+n(2))/(m+n)=0` `:. -3m+2n=0` `:. -3m=-2n` `:. 3m=2n` `:. m/n=(2)/(3)` The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3` Putting `m=2,n=3`, we get the coordinates of point P `y=(my_2+ny_1)/(m+n)` `=(2*6+3*-4)/(2+3)` `=(12-12)/(5)` `=(0)/(5)` Thus, the value of y is `0`
Method-2 : considering the ratio `k:1`Let the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `k:1` The given points are `A(2,-4),B(-3,6)` `:. x_1=2,y_1=-4,x_2=-3,y_2=6` Using section formula `P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))` `:.P(0,y)=((k(-3)+(2))/(k+1),(k(6)+(-4))/(k+1))` `:. (k(-3)+(2))/(k+1)=0` `:. -3k+2=0` `:. -3k=-2` `:. 3k=2` `:. k=(2)/(3)` The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3` Putting `k=2/3`, we get the coordinates of point P `y=(ky_2+y_1)/(k+1)` `=(2/3*6-4)/(2/3+1)` `=(2*6-4*3)/(2+3)` `=(12-12)/(5)` `=(0)/(5)` Thus, the value of y is `0`
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