2. Find the equation of a straight line passing through the points A(7,5) and B(-9,5)
1. Find the equation of a straight line passing through the points `A(7,5)` and `B(-9,5)`
Solution:
The given points are `A(7,5),B(-9,5)`
`:. x_1=7,y_1=5,x_2=-9,y_2=5`
Using two-points formula, The equation of a line AB is
`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`
`:. (y-5)/(5-5)=(x-7)/(-9-7)`
`:. (y-5)/(0)=(x-7)/(-16)`
`:. (y-5)/(0)=(x-7)/(-1)`
`:. -1(y-5)=0(x-7)`
`:. -y +5= +0`
`:. +y-5=0`
Second method :
Points are `A(7,5),B(-9,5)`
`:. x_1=7,y_1=5,x_2=-9,y_2=5`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(5-5)/(-9-7)`
`:. m=(0)/(-16)`
`:.` Slope `=0`
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)=(7,5)` and Slope `m=0` (given)
`:. y-5=0(x-7)`
`:. y -5=0x +0`
`:. +y-5=0`
Hence, The equation of line is `+y-5=0`
2. Find the equation of a straight line passing through the points `A(-1,1)` and `B(2,-4)`
Solution:
The given points are `A(-1,1),B(2,-4)`
`:. x_1=-1,y_1=1,x_2=2,y_2=-4`
Using two-points formula, The equation of a line AB is
`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`
`:. (y-1)/(-4-1)=(x+1)/(2+1)`
`:. (y-1)/(-5)=(x+1)/(3)`
`:. 3(y-1)=-5(x+1)`
`:. 3y -3=-5x -5`
`:. 5x+3y+2=0`
Second method :
Points are `A(-1,1),B(2,-4)`
`:. x_1=-1,y_1=1,x_2=2,y_2=-4`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(-4-1)/(2+1)`
`:. m=(-5)/(3)`
`:.` Slope `=-5/3`
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)=(-1,1)` and Slope `m=-5/3` (given)
`:. y-1=-5/3(x+1)`
`:. 3(y-1)=-5(x+1)`
`:. 3y -3=-5x -5`
`:. 5x+3y+2=0`
Hence, The equation of line is `5x+3y+2=0`
3. Find the equation of a straight line passing through the points `A(-5,-6)` and `B(3,10)`
Solution:
The given points are `A(-5,-6),B(3,10)`
`:. x_1=-5,y_1=-6,x_2=3,y_2=10`
Using two-points formula, The equation of a line AB is
`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`
`:. (y+6)/(10+6)=(x+5)/(3+5)`
`:. (y+6)/(16)=(x+5)/(8)`
`:. (y+6)/(2)=(x+5)/(1)`
`:. (y+6)=2(x+5)`
`:. y +6=2x +10`
`:. 2x-y+4=0`
Second method :
Points are `A(-5,-6),B(3,10)`
`:. x_1=-5,y_1=-6,x_2=3,y_2=10`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(10+6)/(3+5)`
`:. m=(16)/(8)`
`:. m=2`
`:.` Slope `=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,-6)` and Slope `m=2` (given)
`:. y+6=2(x+5)`
`:. y +6=2x +10`
`:. 2x-y+4=0`
Hence, The equation of line is `2x-y+4=0`
4. Find the equation of a straight line passing through the points `A(3,-5)` and `B(4,-8)`
Solution:
The given points are `A(3,-5),B(4,-8)`
`:. x_1=3,y_1=-5,x_2=4,y_2=-8`
Using two-points formula, The equation of a line AB is
`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`
`:. (y+5)/(-8+5)=(x-3)/(4-3)`
`:. (y+5)/(-3)=(x-3)/(1)`
`:. (y+5)=-3(x-3)`
`:. y +5=-3x +9`
`:. 3x+y-4=0`
Second method :
Points are `A(3,-5),B(4,-8)`
`:. x_1=3,y_1=-5,x_2=4,y_2=-8`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(-8+5)/(4-3)`
`:. m=(-3)/(1)`
`:. m=-3`
`:.` Slope `=-3`
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)=(3,-5)` and Slope `m=-3` (given)
`:. y+5=-3(x-3)`
`:. y +5=-3x +9`
`:. 3x+y-4=0`
Hence, The equation of line is `3x+y-4=0`
5. Find the equation of a straight line passing through the points `A(-1,-4)` and `B(3,0)`
Solution:
The given points are `A(-1,-4),B(3,0)`
`:. x_1=-1,y_1=-4,x_2=3,y_2=0`
Using two-points formula, The equation of a line AB is
`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`
`:. (y+4)/(-0+4)=(x+1)/(3+1)`
`:. (y+4)/(4)=(x+1)/(4)`
`:. (y+4)/(1)=(x+1)/(1)`
`:. (y+4)=(x+1)`
`:. y +4=x +1`
`:. x-y-3=0`
Second method :
Points are `A(-1,-4),B(3,0)`
`:. x_1=-1,y_1=-4,x_2=3,y_2=0`
Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`
`:. m=(0+4)/(3+1)`
`:. m=(4)/(4)`
`:. m=1`
`:.` Slope `=1`
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,-4)` and Slope `m=1` (given)
`:. y+4=1(x+1)`
`:. y +4=x +1`
`:. x-y-3=0`
Hence, The equation of line is `x-y-3=0`
This material is intended as a summary. Use your textbook for detail explanation.
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