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)
1. 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)`
Solution:
Given points are `A(1,3)`, `B(3,-5)` and `C(-6,1)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(3,-5)` and `C(-6,1)`
Points are `B(3,-5),C(-6,1)`
`:. x_1=3,y_1=-5,x_2=-6,y_2=1`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(1+5)/(-6-3)`
`:. m=(6)/(-9)`
`:. m=-2/3`
`:.` Slope `=-2/3`
Hence, slope of the line parallel to this line is also `-2/3` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(1,3)` and Slope `m=-2/3` (given)
`:. y-3=-2/3(x-1)`
`:. 3(y-3)=-2(x-1)`
`:. 3y -9=-2x +2`
`:. 2x+3y-11=0`
Hence, The equation of line is `2x+3y-11=0`
2. Find the equation of the line passing through the point `A(4,-5)` and parallel to line passing through the points `B(3,7)` and `C(-2,4)`
Solution:
Given points are `A(4,-5)`, `B(3,7)` and `C(-2,4)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(3,7)` and `C(-2,4)`
Points are `B(3,7),C(-2,4)`
`:. x_1=3,y_1=7,x_2=-2,y_2=4`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(4-7)/(-2-3)`
`:. m=(-3)/(-5)`
`:. m=3/5`
`:.` Slope `=3/5`
Hence, slope of the line parallel to this line is also `3/5` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(4,-5)` and Slope `m=3/5` (given)
`:. y+5=3/5(x-4)`
`:. 5(y+5)=3(x-4)`
`:. 5y +25=3x -12`
`:. 3x-5y-37=0`
Hence, The equation of line is `3x-5y-37=0`
3. Find the equation of the line passing through the point `A(-1,3)` and parallel to line passing through the points `B(0,2)` and `C(4,5)`
Solution:
Given points are `A(-1,3)`, `B(0,2)` and `C(4,5)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(0,2)` and `C(4,5)`
Points are `B(0,2),C(4,5)`
`:. x_1=0,y_1=2,x_2=4,y_2=5`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(5-2)/(4-0)`
`:. m=(3)/(4)`
`:.` Slope `=3/4`
Hence, slope of the line parallel to this line is also `3/4` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(-1,3)` and Slope `m=3/4` (given)
`:. y-3=3/4(x+1)`
`:. 4(y-3)=3(x+1)`
`:. 4y -12=3x +3`
`:. 3x-4y+15=0`
Hence, The equation of line is `3x-4y+15=0`
4. Find the equation of the line passing through the point `A(2,-3)` and parallel to line passing through the points `B(1,2)` and `C(-1,5)`
Solution:
Given points are `A(2,-3)`, `B(1,2)` and `C(-1,5)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(1,2)` and `C(-1,5)`
Points are `B(1,2),C(-1,5)`
`:. x_1=1,y_1=2,x_2=-1,y_2=5`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(5-2)/(-1-1)`
`:. m=(3)/(-2)`
`:. m=-3/2`
`:.` Slope `=-3/2`
Hence, slope of the line parallel to this line is also `-3/2` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(2,-3)` and Slope `m=-3/2` (given)
`:. y+3=-3/2(x-2)`
`:. 2(y+3)=-3(x-2)`
`:. 2y +6=-3x +6`
`:. 3x+2y=0`
Hence, The equation of line is `3x+2y=0`
5. Find the equation of the line passing through the point `A(4,2)` and parallel to line passing through the points `B(1,-1)` and `C(3,2)`
Solution:
Given points are `A(4,2)`, `B(1,-1)` and `C(3,2)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(1,-1)` and `C(3,2)`
Points are `B(1,-1),C(3,2)`
`:. x_1=1,y_1=-1,x_2=3,y_2=2`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(2+1)/(3-1)`
`:. m=(3)/(2)`
`:.` Slope `=3/2`
Hence, slope of the line parallel to this line is also `3/2` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(4,2)` and Slope `m=3/2` (given)
`:. y-2=3/2(x-4)`
`:. 2(y-2)=3(x-4)`
`:. 2y -4=3x -12`
`:. 3x-2y-8=0`
Hence, The equation of line is `3x-2y-8=0`
6. Find the equation of the line passing through the point `A(5,5)` and parallel to line passing through the points `B(1,-2)` and `C(-5,2)`
Solution:
Given points are `A(5,5)`, `B(1,-2)` and `C(-5,2)`
When two lines are parallel, their slopes are equal.
We shall first find the slope of `B(1,-2)` and `C(-5,2)`
Points are `B(1,-2),C(-5,2)`
`:. x_1=1,y_1=-2,x_2=-5,y_2=2`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(2+2)/(-5-1)`
`:. m=(4)/(-6)`
`:. m=-2/3`
`:.` Slope `=-2/3`
Hence, slope of the line parallel to this line is also `-2/3` `(:' m_1=m_2)`
The equation of a line with slope m and passing through `(x_1,y_1)` is `y-y_1=m(x-x_1)`
Here Point `(x_1,y_1)=(5,5)` and Slope `m=-2/3` (given)
`:. y-5=-2/3(x-5)`
`:. 3(y-5)=-2(x-5)`
`:. 3y -15=-2x +10`
`:. 2x+3y-25=0`
Hence, The equation of line is `2x+3y-25=0`
This material is intended as a summary. Use your textbook for detail explanation.
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