(a)Given the points P(3,5),Q(8,10)and R(8,0),find the equation of a line which is perpendicular to QR and passes through the mid point of QR.
(b)The line 2x+3y=2 and 7x-3y=10 intersect at A . Find the equation of the line which passes through A and is perpendicular to the line 3x-4y=7
a) let M(8,5) be the midpoint of QR.
QR is a vertical line, so its slope is undefined.
A line perpendicular to QR must be a horizontal line
A horizontal line through M(8,5) is imply
y = 5
(Where does point P enter the picture ? )
b) Add the two equations ...
9x = 12
x = 12/9 = 4/3
sub into the first ...
8/3 + 3y = 2
3y = -2/3
y = -2/9
point A is (4/3 , -2/9)
The new line is to be perpendicular to
3x - 4y = 7
So it must have the form 4x + 3y = C
but (4/3 , -2/9) lies on it, so
16/3 - 6/9 = c
times 9 ....
48 - 6 = 9c
c = 14/3
we have 4x + 3y = 14/3
or
12x + 9y = 14
(a) To find the equation of a line perpendicular to QR and passing through the midpoint of QR, you can follow these steps:
1. Find the slope of QR: The slope of QR can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of points Q and R respectively.
Slope of QR = (10 - 0) / (8 - 8) = undefined (since the denominator is zero)
2. Since the slope of QR is undefined, we know that QR is a vertical line. To find the equation of a line perpendicular to QR, we need to find a line that is horizontal.
3. Find the midpoint of QR: The midpoint can be found by taking the average of the x-coordinates and the average of the y-coordinates of points Q and R.
Midpoint of QR = ((8 + 8) / 2, (10 + 0) / 2) = (8, 5)
4. Since the line we are looking for is horizontal and passes through the midpoint of QR, we know that the equation will be of the form y = c, where c is the y-coordinate of the midpoint.
5. Therefore, the equation of the line perpendicular to QR and passing through the midpoint of QR is y = 5.
(b) To find the equation of a line that passes through point A and is perpendicular to the line 3x - 4y = 7, you can follow these steps:
1. Convert the given equations to slope-intercept form (y = mx + c), where m is the slope and c is the y-intercept.
2x + 3y = 2 --> 3y = -2x + 2 --> y = (-2/3)x + 2/3
7x - 3y = 10 --> -3y = -7x + 10 --> y = (7/3)x - 10/3
2. Determine the slope of the line 3x - 4y = 7 by rearranging the equation to slope-intercept form.
3x - 4y = 7 --> -4y = -3x + 7 --> y = (3/4)x - 7/4
The slope of the line is (3/4).
3. The line perpendicular to 3x - 4y = 7 will have a negative reciprocal slope. So, the slope of the perpendicular line will be the negative reciprocal of (3/4), which is -(4/3).
4. Since we know the slope of the line (-4/3) and it passes through point A, we can use the slope-intercept form (y = mx + c) with the coordinates of point A to find the equation of the line.
Using point A, (x, y) = (2, 1), we have:
1 = -(4/3)(2) + c
Solve for c:
1 = (-8/3) + c
c = 1 + (8/3)
c = 11/3
5. Therefore, the equation of the line that passes through point A and is perpendicular to the line 3x - 4y = 7 is y = -(4/3)x + 11/3.