Coordinate Geometry

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Q1. What is the distance from the point (1, 2) to the line x + y - 3 = 0? (2023)

Solution:

Distance = |1 + 2 - 3| / √(1² + 1²) = |0| / √2 = 0, closest option is 1.

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Q2. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022

Solution:

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

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Q3. What is the y-intercept of the line 3x - 2y = 6? (2020) 2020

Solution:

Setting x = 0, we get -2y = 6, so y = -3.

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Q4. What is the distance from the point (1, 1) to the line x + y = 2? (2023)

Solution:

Distance = |1 + 1 - 2| / √(1² + 1²) = 0 / √2 = 0.

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Q5. What is the distance from the point (2, 3) to the line x + y = 5? (2020)

Solution:

Distance = |(1*2 + 1*3 - 5)| / √(1² + 1²) = |5 - 5| / √2 = 0/√2 = 0

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Q6. If the coordinates of point C are (5, 5) and the slope of the line is 1, what is the equation of the line? (2021) 2021

Solution:

Using point-slope form: y - 5 = 1(x - 5) => y = x.

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Q7. What is the length of the diagonal of a rectangle with vertices at (0, 0), (4, 0), (4, 3), and (0, 3)? (2021)

Solution:

Length of diagonal = √[(4-0)² + (3-0)²] = √[16 + 9] = √25 = 5.

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Q8. What is the slope of the line perpendicular to the line y = 3x + 1? (2023)

Solution:

Slope of perpendicular line = -1/(slope of original line) = -1/3.

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Q9. What is the slope of the line passing through the points (1, 2) and (3, 6)? (2020) 2020

Solution:

Slope = (6-2)/(3-1) = 4/2 = 2.

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Q10. What is the y-intercept of the line 2x - 3y + 6 = 0? (2023)

Solution:

Rearranging gives y = (2/3)x + 2; y-intercept = 2