To calculate the gradient of line R2, we need to find the negative reciprocal of the gradient of line R1.
a. Gradient of R1: The equation of line R1 is in the form Ax + By + C = 0, where A, B, and C represent the coefficients of x, y, and the constant term, respectively. In this case, A = 2, B = 1, and C = -8. The gradient of R1 can be found by rearranging the equation into y = mx + c form, where m is the gradient.
Given: 2x + y - 8 = 0
Rearranged form: y = -2x + 8
Comparing with y = mx + c, we can see that the gradient of R1 (m1) is -2.
To find the gradient of R2, we take the negative reciprocal of m1:
Gradient of R2 (m2) = -1 / m1
= -1 / (-2)
= 1/2
Therefore, the gradient of R2 is 1/2.
b. Point of intersection (4, k): The point of intersection lies on both R1 and R2. To find k, we need to substitute the x-coordinate (4) into the equation of R1 and solve for y.
Given: R1: 2x + y - 8 = 0
Substituting x = 4 into R1: 2(4) + y - 8 = 0
8 + y - 8 = 0
y = 0
So, at the point (4, k), k = 0.
To find the equation of R2, we use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient.
Given: Point of intersection (4, 0) and gradient of R2 = 1/2
Using the point-slope form for R2, we can substitute the values to find the equation:
y - 0 = (1/2)(x - 4)
y = (1/2)(x - 4)
y = 1/2x - 2
Therefore, the equation of line R2 is y = 1/2x - 2.