Coordinate Geometry - Distance and Section Formula

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Q1. If a circle has a center at (2, -3) and passes through the point (5, -3), what is its radius?

Solution:

Radius = distance from center to point = √((5 - 2)² + (-3 + 3)²) = √(9 + 0) = 3.

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Q2. What is the length of the line segment joining the points (2, 3) and (2, 7)?

Solution:

Length = |y2 - y1| = |7 - 3| = 4.

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Q3. What is the distance from the point (1, 2) to the line 3x + 4y - 12 = 0?

Solution:

Distance = |Ax1 + By1 + C| / √(A² + B²) = |3(1) + 4(2) - 12| / √(3² + 4²) = |3 + 8 - 12| / 5 = | -1 | / 5 = 1/5.

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Q4. What is the equation of a circle with center at (1, 1) and radius 2?

Solution:

The equation of a circle is (x - h)² + (y - k)² = r². Here, r = 2, so (x - 1)² + (y - 1)² = 2² = 4.

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Q5. What is the area of a circle with a diameter of 10?

Solution:

Area = πr². Diameter = 10, so radius r = 5. Area = π(5)² = 25π.

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Q6. What is the radius of a circle with the equation (x - 4)² + (y + 2)² = 36?

Solution:

The standard form of a circle is (x - h)² + (y - k)² = r². Here, r² = 36, so r = √36 = 6.

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Q7. If point A(2, 3) and point B(8, 7) are endpoints of a line segment, what is the midpoint M of AB?

Solution:

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

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Q8. What is the distance between the points (-1, -1) and (3, 3)?

Solution:

Using the distance formula: d = √((3 - (-1))² + (3 - (-1))²) = √(16 + 16) = √32 = 4√2 ≈ 5.66.