Straight Lines

⏱ Time: 0s

Q1. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.

Solution:

Using the formula for distance from a point to a line, d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), we find d = |2(3) + 3(4) - 12| / sqrt(2^2 + 3^2) = 3.

⏱ Time: 0s

Q2. If the line 7x + 2y = 14 is transformed to slope-intercept form, what is the slope?

Solution:

Rearranging to y = -7/2x + 7 shows that the slope is -7/2.

⏱ Time: 0s

Q3. Find the equation of the line that passes through the point (2, -3) and has a slope of 4.

Solution:

Using point-slope form, y + 3 = 4(x - 2) simplifies to y = 4x - 11.

⏱ Time: 0s

Q4. If the line 2x + 3y = 6 intersects the x-axis, what is the point of intersection?

Solution:

Setting y = 0 gives 2x = 6, thus x = 3. The point of intersection is (3, 0).

⏱ Time: 0s

Q5. What is the slope of the line perpendicular to the line 5x + 2y = 10?

Solution:

The slope of the line is -5/2. The slope of the perpendicular line is the negative reciprocal, which is 2/5.

⏱ Time: 0s

Q6. What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

Solution:

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.

⏱ Time: 0s

Q7. What is the equation of the line passing through the points (1, 2) and (3, 4)?

Solution:

The slope m = (4 - 2) / (3 - 1) = 1. Using point-slope form, y - 2 = 1(x - 1) gives y = x + 1.

⏱ Time: 0s

Q8. What is the y-intercept of the line 4x + 5y - 20 = 0?

Solution:

Setting x = 0 in the equation gives 5y = 20, thus y = 4. The y-intercept is 4.

⏱ Time: 0s

Q9. If a line has the equation 7x - 3y = 21, what is its slope?

Solution:

Rearranging to slope-intercept form gives y = (7/3)x - 7. The slope is -7/3.

⏱ Time: 0s

Q10. What is the slope of the line perpendicular to the line 4x + 2y = 8?

Solution:

The slope of the line is -2 (from y = -2x + 4). The slope of the perpendicular line is the negative reciprocal, which is 1/2.