Straight Lines — Practice

Question 1

Find the slope of the line passing through (2, 3) and (5, 9).

Accepted Answer

m = (9-3)/(5-2) = 6/3 = 2. Answer: The slope is 2.

Question 2

Show that the line through (1, 2) and (3, 6) is parallel to the line through (2, -1) and (4, 3).

Accepted Answer

Slope of first line: (6-2)/(3-1)=4/2=2. Slope of second line: (3-(-1))/(4-2)=4/2=2. Since both slopes equal 2, the lines are parallel. Answer: Both lines have slope 2, so they are parallel.

Question 3

Using slopes, check whether the points (1, 2), (3, 6), and (5, 10) are collinear.

Accepted Answer

Slope of the segment from (1,2) to (3,6): (6-2)/(3-1)=2. Slope of the segment from (3,6) to (5,10): (10-6)/(5-3)=2. Since both slopes are equal, the three points are collinear. Answer: Yes, the points are collinear (both slopes equal 2).

Question 4

Find the equation of the line through (-3, 5) with slope -2.

Accepted Answer

y - 5 = -2(x-(-3)) = -2(x+3), so y-5=-2x-6, giving 2x+y+1=0. Answer: The equation is 2x+y+1=0.

Question 5

Find the equation of the line through (2, -3) and (-4, 5).

Accepted Answer

Slope: (5-(-3))/(-4-2)=8/(-6)=-4/3. Using point-slope with (2,-3): y+3=-4/3(x-2), so 3y+9=-4x+8, giving 4x+3y+1=0. Answer: The equation is 4x+3y+1=0.

Question 6

Write the equation of the line with slope 3 and y-intercept -2.

Accepted Answer

Using y=mx+c with m=3, c=-2: y=3x-2. Answer: The equation is y=3x-2.

Question 7

Find the equation of the line with x-intercept 5 and y-intercept -3.

Accepted Answer

Using x/a+y/b=1 with a=5, b=-3: x/5 - y/3 = 1. Multiplying through by 15: 3x-5y=15, i.e. 3x-5y-15=0. Answer: The equation is 3x-5y-15=0.

Question 8

Find the acute angle between two lines with slopes 2 and -3.

Accepted Answer

tan(theta) = |(2-(-3))/(1+2(-3))| = |5/(1-6)| = |5/(-5)| = 1. So theta = 45 degrees. Answer: The acute angle between the lines is 45 degrees.

Question 9

Find the perpendicular distance from the point (2, 3) to the line 3x - 4y + 5 = 0.

Accepted Answer

d = |3(2)-4(3)+5| / sqrt(3^2+(-4)^2) = |6-12+5|/5 = |-1|/5 = 1/5. Answer: The distance is 1/5.

Question 10

Find the distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0.

Accepted Answer

Here A=3, B=4, C1=-5, C2=10. d=|C1-C2|/sqrt(A^2+B^2)=|-5-10|/sqrt(9+16)=15/5=3. Answer: The distance between the lines is 3.

Practice — Straight Lines

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