Learning these TS Inter 2nd Year Maths 2B Formulas Chapter 2 System of Circles will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2B System of Circles Formulas

→ We denote circles by S = x^{2} + y^{2} + 2gx + 2fy + c = 0 and S’ = x^{2} + y^{2} + 2g’x + 2f’y + c’ = 0.

→ If C_{1}, C_{2} are the centres and r_{1}, r_{2} are radii of two intersecting circles S = 0 and S’ = 0 and C_{1}C_{2} = d then if θ is the angle between them, then cos θ = \(\frac{d^2-r_1^2-r_2^2}{2 r_1 r_2}\)

→ If 0 is the angle between two intersecting circles S = 0 and S’ = 0, then

cos θ = \(\frac{c+c^{\prime}-2 g g^{\prime}-2 f^{\prime}}{2 \sqrt{g^2+f^2-c} \sqrt{g^2+f^2-c^{\prime}}}\)

→ Two circles S = 0 and S’ = 0 are orthogonal ⇔ 2(gg’ + ff’) = c + c’.

→ If S = 0, S’ = 0 are any two intersecting circles and λ, µ are two real numbers such that λ + µ ≠ 0 and λS + µS’ = 0 represents a circle passing through the intersection of circles S = 0, S’ = 0.

→ If S = 0, S’ = 0 are any two intersecting circles and ke R (≠ -1) then S + kS’ = 0 represents a circle passing through their point of intersection.

→ If S = 0 and a straight line L = 0 intersect, then for any real number k, S + kL = 0 represents a circle passing through their intersection.

→ The equation of common chord of two intersecting circles S = 0, S’ = 0 is S – S’ = 0.

→ The equation of common tangent at the point of contact when the circles S = 0, S’ = 0 touch each other is S – S’ = 0.

→ The radical axis of two circles is defined to be the locus of a point which moves such that its powers with respect to the two given circles are equal.

→ The radical axis of two circles S = 0, S’ = 0 is

- the common chord when the two circles intersect at two distinct points.
- the common tangent at the point of contact when the circles touch each other.

→ The radical axis of any two circles bisects the line segment joining the points of contact of the common tangent of these two circles.