Coordinate geometry modelling

Objective. Translate geometric constraints into a connected coordinate model.

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2. Worked example
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Learn: the key idea

Coordinate geometry modelling means translating a geometric description, such as points forming a triangle or circle, into equations using distance, midpoint and gradient formulae, then solving for an unknown. For example, if a circle must pass through (0,0) and have centre on the line y=x, write the centre as (a,a), then equate the radius² using both the origin and a second given point to find a. This approach turns a wordy geometric constraint into solvable algebra. Always finish by checking the found values satisfy every original condition stated in the problem.

Worked example

A circle has centre on the x-axis and passes through (0,3) and (4,3). Find the circle's centre.

1 $Let the centre be (a,0) since it lies on the x-axis$
2 $Equal radii give a²+9 = (a-4)²+9$
3 $Expand: a² = a²-8a+16, so 8a=16$
4 $Solve to get a=2, so the centre is (2,0)$

Answer: (2, 0)

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Common mistake. Students mix up which point is the centre and which is a point on the circle, substituting incorrectly into the distance formula; label given points clearly before writing any equation.

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