Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Find the circle through O=(0,0), A=(2t,0), B=(0,2).

Read the idea, work independently, then explain what changed.

Return to the lesson / paper ↗

TOPIC 01

Equations of lines and circles: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Find the circle through O=(0,0), A=(2t,0), B=(0,2).

t=28t=28
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
F=0;D=−2t;E=−2F=0;\quad D=-2t;\quad E=-2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3136+56D=0⇒D=−563136+56D=0\Rightarrow D=-56

  3. Apply the stated relation and retain its conditions.

    4+2E=0⇒E=−24+2E=0\Rightarrow E=-2

  4. Three noncollinear points determine one circle.

The requested relation or conclusion is shown below.

x2+y2−56x−2y=0x^2+y^2-56x-2y=0

Checks and common pitfalls: Three noncollinear points determine one circle.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗