Question 1025978: 7. Consider the line L represented by the equation x + 4y = 12. (a) Find the equation of the line which is parallel to L, and passes through (1,−2). (b) Find the equation of the line which is perpendicular to L, and passes through (0,1). (c) Find the intersection point of L and the line 2x−3y + 9 = 0 (d) Find the perpendicular distance from the point P(−1.5) to
Answer by Cromlix(4381)   (Show Source):
You can put this solution on YOUR website! Hi there,
a) Sort equation into y = mx + c form.
x + 4y = 12
4y = -x + 12
y = -1/4x + 3
Gradient (Slope) = -1/4
Lines that are parallel have the
same gradient.
m1 = m2
Using line equation
y - b = m(x - a)
m = -1/4 (a,b) = 1,-2)
y - (-2) =-1/4(x - 1)
y + 2 = -1/4x + 1/4
y = -1/4x + 1/4 - 8/4 (2)
y = -1/4x - 7/4
b) Lines that are perpendicular to
one another have gradients that
multiply together to give -1
m1 x m2 = -1
-1/4 x m2 = -1
m2 = 4
Using line equation
y - b = m(x - a)
m = 4 (a,b) = (0,1)
y - 1) = 4(x - 0)
y - 1 = 4x
y = 4x + 1
c)Simultaneous equation:
x + 4y = 12......(1)
2x - 3y = -9.....(2)
Multiply Eq(1) by 2
2x + 8y = 24......(1)
2x - 3y = -9......(2)
Subtract Eq(2) from Eq(1)
11y = 33
y = 3
Substitute y = 3
into Eq(2)
2x - 3y = -9
2x - 3(3) = -9
2x - 9 = -9
2x = -9 + 9
2x = 0
x = 0
{0.3} Intersection point.
d) not sure what the other point
was as it was cut off.
Assuming the intersection point.
(0,3) and P(-1,5)
Formula:
√(x2 - x1)^2 + (y^2 - y^1)^2
√(-1 - 0)^2 + (5 - 3)^2
√ 1 + 4
= √5
= 2.236 units
Hope this helps :-)
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