Concept understanding — Circle Equation Standard Form
Where the Circle Equation Comes From
Imagine you're standing at a point on a flat field. You tie a rope to a stake at that point, walk out the full length of the rope, and start walking in a circle, keeping the rope taut. Every point you step on is exactly the same distance from the stake.
That's the entire idea: a circle is the set of all points that are a fixed distance (the radius) from a fixed point (the centre).
If we put this on a coordinate plane, we can turn that geometric idea into an algebraic equation.
From Geometry to Algebra
Let the centre be at coordinates (h,k). Let the radius be r. Take any point (x,y) that lies on the circle. The distance from (x,y) to (h,k) must equal r.
What's the distance between two points in the plane? The distance formula:
(x−h)2+(y−k)2=r
Now square both sides to remove the square root:
(x−h)2+(y−k)2=r2
That's it. That's the standard form of the equation of a circle.
(x−h)2+(y−k)2=r2
where (h,k) is the centre and r is the radius (r>0).
What Each Piece Tells You
(x−h) and (y−k) — these shift the circle away from the origin. If the centre is at (0,0), the equation simplifies to x2+y2=r2.
r2 — notice it's the square of the radius, not the radius itself. If the equation says x2+y2=25, the radius is 25=5, not 25.
The equals sign — only the points (x,y) that make this equation true lie on the circle. Any other point gives a larger or smaller left-hand side.
Watch out
A common mistake: for (x−3)2+(y+2)2=16, students often read the centre straight off the signs printed in the equation and say (3,2). That's wrong. Each bracket must first be written in the exact form x−h and y−k: here (y+2)=(y−(−2)), so k=−2, not 2. The centre is actually (3,−2). Always flip the sign inside every bracket before reading off h and k.
Quick Example
Write the equation of a circle with centre (−1,4) and radius 3.
Here h=−1, k=4, r=3. Plug in:
(x−(−1))2+(y−4)2=32
Simplify:
(x+1)2+(y−4)2=9
That's the standard form. From this, you can immediately read off the centre (−1,4) and radius 3.
Why This Form Matters
The standard form is the most useful because it gives you the centre and radius at a glance. In exams, you'll often be given an expanded form like x2+y2−6x+4y−12=0 and asked to rewrite it in standard form by completing the square — that's the next step in your learning, but the standard form itself is the destination.
For now: centre tells you where, radius tells you how big, and the equation tells you which points belong.
The Standard Form of a Circle's Equation is one of the first results in the NCERT Class 11 Mathematics chapter on Conic Sections, matching searches like "equation of a circle: definition, formula and examples" or "conic sections important questions class 11 maths". Recognising centre and radius directly from this form is also a routine, quick-scoring question type in CBSE boards, JEE Main, and state CET coordinate geometry sections.
Concept: Circle Equation Standard Form — (x−h)2+(y−k)2=r2.
Step 1: Identify centre (h,k)=(21,41) and radius r=121.
Step 2: Substitute into the standard form:
(x−21)2+(y−41)2=(121)2
Step 3: Square the radius:
(121)2=1441
✓Final answer
The equation is (x−21)2+(y−41)2=1441.
The standard form of a circle is (x−h)2+(y−k)2=r2. Substituting the given centre (21,41) and radius 121, then simplifying, gives the equation 36x2+36y2−36x−18y+11=0.
The equation of any circle is built from its centre and radius. If you know the centre (h,k) and the radius r, the circle is the set of all points (x,y) that are exactly r units away from (h,k). That distance condition is just the Pythagorean theorem in disguise.
(x−h)2+(y−k)2=r2
This is the standard form of a circle. It’s the most direct way to write the equation when you’re given the centre and radius. No shifting, no completing the square — just plug in and simplify.
Substitute the centre and radius.
Here h=21, k=41, and r=121.
(x−21)2+(y−41)2=(121)2
Square the radius.(121)2=1441. So we have:
(x−21)2+(y−41)2=1441
Expand the squares.
(x−21)2=x2−x+41
(y−41)2=y2−21y+161
Adding them:
x2+y2−x−21y+41+161=1441
Combine the constant terms.41=164, so 41+161=165.
The equation becomes:
x2+y2−x−21y+165=1441
Move the constant to the right side.
x2+y2−x−21y=1441−165
Compute the right side. 165=14445, so:
1441−14445=−14444=−3611
Thus:
x2+y2−x−21y=−3611
Clear the fractions by multiplying through by 36.
36x2+36y2−36x−18y=−11
Bring everything to one side.
36x2+36y2−36x−18y+11=0
Watch out
A common mistake is forgetting to square the radius, or mishandling the fractions when combining 41 and 161. Always write every term with a common denominator before adding or subtracting — it saves errors.
Tip
If you prefer to avoid fractions entirely, multiply the standard form by the least common multiple of the denominators (here 144) right after substitution. That gives integer coefficients from the start, though the expansion is a bit heavier. The method above keeps the algebra cleaner step by step.
✓Final answer
The equation of the circle is 36x2+36y2−36x−18y+11=0.