CircleJEE Main

Concept

A circle is the set of points a fixed distance $r$ (the radius) from a fixed point $(h,k)$ (the centre). Its standard equation falls straight out of the distance formula.

Key formula

Derivation

Any point $(x,y)$ on the circle is distance $r$ from $(h,k)$: $\sqrt{(x-h)^2+(y-k)^2} = r$. Squaring gives the standard form.

Expanding yields the general form $x^2+y^2+2gx+2fy+c=0$ with centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$. The tangent length from an external point at distance $d$ is $\sqrt{d^2-r^2}$.

Scenarios to explore

• Circle — Centre, radius, equation and tangents from a point.

Real-world applications

• Locus problems and circular motion paths.
• Range circles in navigation and GPS trilateration.
• Defining arcs and fillets in design.

JEE exam tips

• A line is tangent when its distance from the centre equals $r$.
• Power of a point $= d^2 - r^2$: positive outside, zero on, negative inside.

Common mistakes

• Sign error: centre of $x^2+y^2+2gx+2fy+c=0$ is $(-g,-f)$, not $(g,f)$.
• Forgetting to square the radius on the right-hand side.
• Taking a tangent length from a point inside the circle (it doesn't exist).

Exam traps to avoid

• $g^2+f^2-c$ must be positive for a real circle; otherwise it's a point or imaginary.
• The origin being inside the circle means no real tangent can be drawn from it.