Lesson 3 of 12 · 8 min
The circle
NCERT §10.3
The club marks a round flower bed on the terrace, which is gridded in metres with a water tap at the origin. The plan says x² + y² − 6x − 8y = 0. Where is the centre, and does the bed reach the tap?
The lesson in notes
In short
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.