Q1. Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.
Solution:
The slope of the given line is 3, so the slope of the perpendicular line is -1/3. Using point-slope form, we get y + 1 = -1/3(x - 4), which simplifies to y = -1/3x + 11/3.
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Q2. What is the x-intercept of the line 5x - 2y + 10 = 0?
Solution:
Setting y = 0 in the equation gives 5x + 10 = 0, thus x = -2.
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Q3. 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.
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Q4. Find the equation of the line parallel to y = 3x + 2 that passes through the point (4, 1).
Solution:
Since the line is parallel, it has the same slope (3). Using point-slope form: y - 1 = 3(x - 4) gives y = 3x - 11.
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Q5. Find the x-intercept of the line 5x - 2y + 10 = 0.
Solution:
Setting y = 0 in the equation gives 5x + 10 = 0, thus x = -2. The x-intercept is -2.
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Q6. If the line 3x - 4y + 12 = 0 is parallel to which of the following lines?
Solution:
Parallel lines have the same slope. The slope of the given line is 3/4, which is the same as the slope of 6x - 8y + 24 = 0.
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Q7. If a line has the equation 7x - 3y + 21 = 0, what is its slope?
Solution:
Rearranging gives y = (7/3)x + 7. The slope is -7/3.
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Q8. 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.
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Q9. What is the distance between the points (2, 3) and (5, 7)?
Solution:
Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.
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Q10. Which of the following points lies on the line 4x - 5y + 20 = 0?
Solution:
Substituting (0, -4) into the equation gives 4(0) - 5(-4) + 20 = 0, which is true.
