4A: 2: Equations of Straight Lines

Question 1

Find the distance between points (x, y) and (y, x).

Answer 1

Distance = √[(x-y)² + (y-x)²] = (√2)(y-x)².

Question 2

Find the slope of the line segment connecting points (x, y) and (y, x).

Answer 2

slope = (x-y)/(y-x) = -1

Question 3

Find the inclination of a line L₁ connecting the origin and point (3, √3). Hence determine if the line L₂ connecting (3, √3) and (6, √6) passes through the origin.

Answer 3

Inclination of L₁ = tan^-1[(√3)/3] = 30º
Slope of L₂ = (√6-√3)/(6-3) = (√3)(√2 - 1)/3 ≠ √3/3
Since their slopes are not equal, L₂ does not pass throught the origin.

Question 4

(Ch2 notes)
Given A (2, 5), B (-6, -3), and Q are collinear such that AB = 4QB. Find the coordinates of Q.

Answer 4

Case 1: Q lies on the segment AB.
AQ:QB = 3:1
x-coordinate of Q = [3(-6)+1(2)]/(3+1) = -4
y-coordinate of Q = [3(-3)+1(5)]/(3+1) = -1
coordinates of Q = (-4, -1)
Case 2: Q does not lie on the segment AB.
AB:BQ = 4:1
Let the coordinates of Q be (x, y).
Solving (4x+2)/(4+1) = -6 and (4y+5)/(4+1) = -3,
x = -8, y = -5
The coordinates of Q are (-8, -5).

Question 5

Given that the centroid divides a median of a triangle into two segments of length 2:1, derive the coordinates of the centroid of a triangle if the coordinates of its vertices are A (a, b), B (c, d) and C (e, f).

Answer 5

Let M be the mid-point of AB and G be the centroid.
Coordinates of M = ([a+c]/2, [b+d]/2)
Coordinates of G = ({2[a+c]/2+e}/[2+1],
{2[b+d]/2+f}/[2+1]) = ([a+c+e]/3, [b+d+f]/3)

Question 6

Find the equation of the ine perpendicular to y=4 and passing through the origin.

Answer 6

vertical line: x=0

Question 7

Find the equation of the straight line L passing through (-1, 4) and with inclination 135º. Express it in slope-intercept form.

Answer 7

Slope = tan(135º) = -1
equation: y-4 = -1(x+1)
y-4 = -x-1
y = -x+3

Question 8

Prove that for a straight line having equation x/a+y/b=1, the x-intercept and y-intercepts of thie line are a and b respectively.

Answer 8

x/a+y/b = 1
bx+ay = ab
ay = -bx+ab
y = -b/a x+b
y-intercept: b
x-intercept: sub y=0, (-b/a)x = -b, x=a
x-intercept: a

Question 9

For the straight line ax+by+c=0, express its slope in terms of a, b, and c.

Answer 9

-a/b

Question 10

For the straight line ax+by+c=0, express its x-intercept in terms of a, b, and c.

Answer 10

-c/a

Question 11

For the straight line ax+by+c=0, express its y-intercept in terms of a, b, and c.

Answer 11

-c/b