1. The equation, in general form, of the line that passes through the point (3,4) and is perpendicular to the line 5 x + 5 y + 9 = 0 is Ax + By + C = 0, where a=? b=? c=?
2.The equation of the line that goes through the points ( -9 ,-6 ) and ( -6 ,-3 ) can be written in general form A x + B y + C = 0 where B=?
1.
The line perpendicular to the given line
Ax + By + C = 0
is
Bx - Ay + D = 0
where D has to be found from an intercept or a given point through which the new line passes.
For the given line
5x + 5y + 9 = 0
the required line is
5x - 5y + D = 0
Since it passes through (3,4)
D can be found by substituting x=3 and y=4
5*3 -5*4 + D = 0
D = 5
Thus the required line is
5x -5y + 5 = 0
check: 5(3)-5(4)+5 =15-20+5=0 OK
2.
A line passing through two given points P1(x1,y1) and P2(x2,y2) is given by
(y2-y1)(x-x1) - (x2-x1)(y-y1) =0
substitute P1(-9,-6), P2(-6,-3) to get
(-3-(-6)(x-(-9)) - (-6-(-9))(y-(-6))=0
or
x-y+3=0
after dividing both sides of the equal sign by 3.
A=1, B=-1, C=3.
Check
-9-(-6)+3=0
-6-(-3)+3=0
OK.
1. To find the equation of a line that is perpendicular to another line, we need to know that the slopes of the two lines are negative reciprocals of each other.
First, let's find the slope of the given line. The equation 5x + 5y + 9 = 0 can be rewritten as 5y = -5x - 9 by isolating the y-term. Dividing both sides by 5, we get y = -x - 9/5, which is in slope-intercept form (y = mx + b), where m is the slope of the line.
The slope of the given line is -1.
Since the line we're looking for is perpendicular, its slope will be the negative reciprocal of -1. In other words, the slope of the line we want is 1.
Now we have the slope (m = 1) and the point (3, 4) that the line passes through. We can use the point-slope form of a line to write the equation:
y - y1 = m(x - x1), where (x1, y1) is a given point and m is the slope.
Plugging in the values, we get:
y - 4 = 1(x - 3)
Expanding and simplifying:
y - 4 = x - 3
Rearranging to fit the general form (Ax + By + C = 0):
x - y - 1 = 0
Therefore, A = 1, B = -1, and C = -1.
2. To find the equation of a line that passes through two given points, we can use the point-slope form or the slope-intercept form of a line.
Let's use the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is a given point and m is the slope.
Given points: (-9, -6) and (-6, -3)
First, let's find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1)
m = (-3 - (-6)) / (-6 - (-9))
= (-3 + 6) / (-6 + 9)
= 3 / 3
= 1
Now we have the slope (m = 1) and the point (-9, -6) that the line passes through.
Using the point-slope form:
y - (-6) = 1(x - (-9))
Expanding and simplifying:
y + 6 = x + 9
Rearranging to fit the general form (Ax + By + C = 0):
x - y - 3 = 0
Therefore, B = -1.