Conic Sections

Maths · Class 11

Simulation · Maths · Class 11

Complete the square, find the circle

"Complete the square, find the circle" is a free, interactive Class 11 Maths simulation for JEE, built on the NCERT section "The circle" in Conic Sections. X² + y² + Dx + Ey + F = 0 with sliders; the completed-square form, centre (−D/2, −E/2) and r² are shown, and the circle disappears when r² < 0. JEE Main asked 116 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson The circle in Conic Sections. Change the values and watch what happens.

Complete the square, find the circleMaths · Class 11

What it shows

NCERT §10.3

X² + y² + Dx + Ey + F = 0 with sliders; the completed-square form, centre (−D/2, −E/2) and r² are shown, and the circle disappears when r² < 0.

Controls

3 to try

  • SliderD −12 to 12
  • SliderE −12 to 12
  • SliderF −60 to 40

Try this

Produce the circle with centre (−4, −5) and radius 7.

Common mistake

Completing the square with the wrong constant on the right.

In JEE Main

42 shifts analysed

116 questions from Conic Sections across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §10.3

  • A circle is the set of all points of a plane at the same distance from one fixed point of that plane. The fixed point is the centre and the common distance is the radius.
  • If the centre is C(h, k) and the radius is r, a point P(x, y) lies on the circle exactly when CP = r. Squaring the distance formula gives (x − h)² + (y − k)² = r².
  • With the centre at the origin this becomes x² + y² = r². The circle with centre (−3, 2) and radius 4 is (x + 3)² + (y − 2)² = 16.
  • To read the centre and radius from an expanded equation, complete the squares. x² + y² + 8x + 10y − 8 = 0 becomes (x + 4)² + (y + 5)² = 8 + 16 + 25 = 49, so the centre is (−4, −5) and the radius is 7.
  • A circle needs three facts to fix h, k and r. For the circle through (2, −2) and (3, 4) with its centre on x + y = 2, equating the two squared distances and using h + k = 2 gives h = 0.7, k = 1.3 and r² = 12.58.
  • A point is inside, on or outside a circle according as its distance from the centre is less than, equal to or greater than r; comparing the squares avoids square roots. For x² + y² = 25, the point (3, −3.5) gives 9 + 12.25 = 21.25 < 25, so it lies inside.

Questions

4 answers

What does the "Complete the square, find the circle" simulation show?
X² + y² + Dx + Ey + F = 0 with sliders; the completed-square form, centre (−D/2, −E/2) and r² are shown, and the circle disappears when r² < 0.
What mistake does the "Complete the square, find the circle" simulation fix?
A common mistake it clears up: completing the square with the wrong constant on the right.
Does JEE Main ask questions from Conic Sections?
Yes. JEE Main asked 116 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "The circle" in Conic Sections (Class 11 Maths).

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