Families of straight lines

Question

A family of lines have equations of the form 3x + by = 12, where b is a non-zero

number.

a) Find the y-axis intercept of a line in this family in terms of b.

b) Find the gradient of the line in terms of b.

c) Find the value of b if the gradient is

i 1

ii −2

d) Find the equation of the line perpendicular to the line 3x + by = 12 at the point (4, 0).

Answer 1

3x + by = 12

a) The y-intercept occurs when x= 0. So plug x = 0 into the equation and solve for y - that'll be the y-intercept.

b) Put the equation into the slope-intercept form, y = mx + c, by solving for y. The slope m is the gradient.

c) (i) Set the slope value from b (above) equal to 1 and solve for b

(ii) Set the slope value from b above equal to -2 and solve for b

d) The equation of the perpendicular line also has the form y = mx + c, but the slope is equal to the negative reciprocal of the slope value you got in part b above. So if that value is -2, then the slope of the perpendicular line is -(1/(-2) = 1/2. Plug that value of the slope into y = mx + c. Then plug in the x and y values from the given point, (4,0), and solve for c.