A straight line passes through points (8, -2) and (4, -4)
a)Write its equation in the form ax + by + c = 0 where a, b and c are integers.
b)If the line in (a) above cuts the x axis at point P, determine the coordinates of P.
c)Another line which is perpendicular to the line in (a) above passes through point P and cuts the
Y axis at the point Q. Determine the co-ordinates of point Q.
d)Find the length of QP.
a) First, let's find the slope of the line passing through points (8, -2) and (4, -4):
Slope (m) = (y2 - y1) / (x2 - x1)
= (-4 - (-2)) / (4 - 8)
= (-2) / (-4)
= 1/2
Now, we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
y - (-2) = 1/2(x - 8)
y + 2 = 1/2x - 4
1/2x - y - 6 = 0
Multiplying by 2 to clear fractions:
x - 2y - 12 = 0
Therefore, the equation of the line is x - 2y - 12 = 0
b) To find where the line intersects the x-axis, we set y = 0 in the equation:
x - 2(0) - 12 = 0
x - 12 = 0
x = 12
So, the line intersects the x-axis at point P(12, 0).
c) Since the line in (a) is perpendicular to the new line we are looking for, their slopes are negative reciprocals of each other.
Slope of line in (a) = 1/2
Slope of perpendicular line = -2
Now, we have a point (12, 0) and the slope (-2) to find the equation of the new line passing through point P:
y - 0 = -2(x - 12)
y = -2x + 24
To find the coordinates of point Q where the line cuts the y-axis, we set x = 0:
y = -2(0) + 24
y = 24
Q(0, 24)
d) To find the length of QP, we use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
distance = sqrt((0 - 12)^2 + (24 - 0)^2)
distance = sqrt((-12)^2 + 24^2)
distance = sqrt(144 + 576)
distance = sqrt(720)
distance ≈ 26.83
Therefore, the length of QP is approximately 26.83.