What are the coordinates of the centre of the circle ?

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Always double-check the arithmetic when averaging the coordinates. A small mistake here will lead to incorrect answers for the radius and equation of the circle as well.

Updated On: Mar 16, 2026

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Solution and Explanation

Given the endpoints of a diameter, we need to find the center of the circle.

The center is the midpoint of the diameter. We use the midpoint formula:
Midpoint = ( (x₁ + x₂)/(2), (y₁ + y₂)/(2) ) The endpoints are (x₁, y₁) = (3, 5) and (x₂, y₂) = (5, 9).
Center (xc, yc):
xc = (3 + 5)/(2) = (8)/(2) = 4 yc = (5 + 9)/(2) = (14)/(2) = 7 The coordinates of the center are (4, 7).

The coordinates of the centre of the circle are (4, 7).

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Question: 2

What is its radius ?

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Using the distance formula between the center and an endpoint is often simpler as the numbers might be smaller. However, calculating the full diameter length and halving it is a good way to verify your answer.

Updated On: Mar 16, 2026

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Solution and Explanation

We need to find the radius of the same circle.

The radius is the distance from the center to any point on the circle. We can calculate the distance between the center and one of the given endpoints of the diameter.
Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²).
Alternatively, we can find the length of the diameter and divide by 2.

Method 1: Distance from center to endpoint.
Center: (4, 7). Endpoint: (3, 5).
Radius r = √((4-3)² + (7-5)²)
r = √((1)² + (2)²) r = √(1 + 4) = √(5) Method 2: Half the length of the diameter.
Endpoints of diameter: (3, 5) and (5, 9).
Length of diameter d = √((5-3)² + (9-5)²)
d = √((2)² + (4)²) d = √(4 + 16) = √(20) Radius r = (d)/(2) = √(20)2 = √(4 × 5)2 = 2√(5)2 = √(5)
Both methods give the same result.

The radius is √(5).

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Question: 3

write the equation of the circle.

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A common mistake is forgetting to square the radius for the right-hand side of the equation. Remember the formula is (x-h)² + (y-k)² = r², not r.

Updated On: Mar 16, 2026

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Solution and Explanation

We need to write the standard equation of the circle using the center and radius we've found.

The standard equation of a circle with center (h, k) and radius r is:
(x-h)² + (y-k)² = r² From the previous parts, we have:
Center (h, k) = (4, 7).
Radius r = √(5).
Therefore, r² = (√(5))² = 5.
Substituting these values into the standard equation:
(x-4)² + (y-7)² = 5 The equation of the circle is (x-4)² + (y-7)² = 5.

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Question: 4

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

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Slope is often described as "rise over run". The "rise" is the change in y (Δ y), and the "run" is the change in x (Δ x). It doesn't matter which point you choose as (x₁, y₁) as long as you are consistent. (5-9)/(3-5) = (-4)/(-2) = 2, which gives the same result.

Updated On: Mar 16, 2026

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Solution and Explanation

This is an alternative question. We need to find the slope of the line segment that was the diameter in the previous question.

The formula for the slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ - y₁)/(x₂ - x₁) The two given points are (x₁, y₁) = (3, 5) and (x₂, y₂) = (5, 9).
Substituting these values into the slope formula:
m = (9 - 5)/(5 - 3) m = (4)/(2) m = 2 The slope of the line passing through the points is 2.

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