Simulation · Maths · Class 11
Complete the square, find the circle
From the lesson The circle in Conic Sections. Change the values and watch what happens.
Complete the square, find the circleMaths · Class 11
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.