A line L1 passes through point (1, 2) and has gradient of 5. Another line L2, is perpendicular to L2 and meets it at a point where x = 4. Find the equation for L2 in the form of y = mx + c
L1:
y-2 = 5(x-1)
y-2 = 5x-5
5x - y = 3
L2: x + 5y = c
from L1, when x = 4, y = 17
so (4,17) into L2
4 + 85 = c = 89
L2: x + 5y = 89 or y = (-1/5)x + 89/5
The line l1 has equation 3x + 5y − 7 = 0
(a) Find the gradient of l1
(2)
The line l2 is perpendicular to l1 and passes through the point (6, −2).
(b) Find the equation of l2 in the form y = mx + c, where m and c are constants.
The point A has coordinates (−4, 11) and the point B has coordinates (8, 2).
(a) Find the gradient of the line AB, giving your answer as a fully simplified fraction.
(2)
The point M is the midpoint of AB. The line l passes through M and is perpendicular to AB.
(b) Find an equation for l, giving your answer in the form px + qy + r = 0 where p, q and r are integers to be found.
(4)
The point C lies on l such that the area of triangle ABC is 37.5 square units.
(c) Find the two possible pairs of coordinates of point C.
Gradient=(Y2-Y1)/(X2-X1)
5=(y-2)/(4-1)
y=17
gradient of L1×gradient of L2=-1
5×L2=-1
Gradient of L2=-1/5
5(y-17)=-1(x-1)
y=-1/5x+17 1/5
To find the equation for line L2, we need to first determine its gradient. Given that L2 is perpendicular to L1, we know that the product of their gradients will be -1.
The gradient of L1 is 5, so we can calculate the gradient of L2 by taking the negative reciprocal of 5:
Gradient of L2 = -1/5
We also know that L2 passes through the point where x = 4. Let's call this point (4, y).
Now we can use the point-slope form of a linear equation (y - y1) = m(x - x1) to find the equation for L2:
(y - 2) = (-1/5)(x - 4)
Next, let's simplify the equation by distributing -1/5 to (x - 4):
y - 2 = (-1/5)x + (4/5)
To isolate y, let's add 2 to both sides of the equation:
y = (-1/5)x + (4/5) + 2
Simplifying further:
y = (-1/5)x + (4/5) + 10/5
y = (-1/5)x + (14/5)
Therefore, the equation for line L2 in the form of y = mx + c is:
y = (-1/5)x + 14/5
On L1, as x increases by 3, y increases by 5*3, so (4,17) is on L1
The perpendicular line has slope -1/5, so L2 is
y-17 = -1/5 (x-4)
Now you can convert that to slope-intercept form.