Circles

The circle is one of the most important conic sections in JEE Mathematics. A circle is the locus of a point moving at a constant distance (radius) from a fixed point (centre). Its study in coordinate geometry involves equations, tangents, normals, chord properties, and interactions with other circles and lines.

The standard form of a circle with centre (h, k) and radius r is (x - h)² + (y - k)² = $r^{2}$.
Expanding this gives the general form $x^{2}$ + $y^{2}$ + 2gx + 2fy + c = 0, where centre = (-g, -f) and radius = $\sqrt{g^{2} + f^{2} - c}$. For the circle to be real, $g^{2}$ + $f^{2}$ - c > 0.
When $g^{2}$ + $f^{2}$ = c, it degenerates to a point circle; when $g^{2}$ + $f^{2}$ < c, the circle is imaginary.

The position of a point P(x1, y1) relative to a circle $x^{2}$ + $y^{2}$ + 2gx + 2fy + c = 0 is determined by S₁ = $x1^{2}$ + $y1^{2}$ + 2gx1 + 2fy1 + c.
If S₁ < 0, the point is inside; S₁ = 0, on the circle; S₁ > 0, outside. The length of the tangent from an external point is $\sqrt{S1}$.

A line y = mx + c intersects the circle $x^{2}$ + $y^{2}$ = $a^{2}$ when |c|/$\sqrt{1 + m^{2}}$ < a, is tangent when |c|/$\sqrt{1 + m^{2}}$ = a, and does not intersect when |c|/$\sqrt{1 + m^{2}}$ > a.
The tangent at point (x1, y1) on $x^{2}$ + $y^{2}$ = $a^{2}$ is xx1 + yy1 = $a^{2}$ (obtained by the T = 0 formula).

The equation of the tangent to $x^{2}$ + $y^{2}$ = $a^{2}$ with slope m is y = mx +/- a*$\sqrt{1 + m^{2}}$.
The chord of contact from an external point (x1, y1) is T = 0, i.e., xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.

Two circles S₁ = 0 and S₂ = 0 intersect at two points when |r1 - r2| < d < r1 + r2, where d is the distance between centres. They are tangent externally when d = r1 + r2 and internally when d = |r1 - r2|. The radical axis of two circles S₁ - S₂ = 0 is the locus of points having equal power with respect to both circles. The family of circles through the intersection of S₁ and S₂ is S₁ + $\lambda$*S₂ = 0.

The key problem-solving concept is mastering the T = 0 substitution (replacing $x^{2}$ with xx1, $y^{2}$ with yy1, x with (x+x1)/2, y with (y+y1)/2) which unifies tangent, chord of contact, chord with midpoint, and pair of tangents formulas.