Learning these TS Inter 2nd Year Maths 2B Formulas Chapter 1 Circles will help students to solve mathematical problems quickly.
TS Inter 2nd Year Maths 2B Circles Formulas
→ The locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle. The fixed point is called “centre” and the fixed distance is called the “radius” of the circle.
→ The equation of a circle with centre (h, k) and radius ‘r’ is (x – h)2 + (y – k)2 = r2.
→ The equation of a circle in standard form is x2 + y2 = r2 where (0, 0) is the centre and ‘r’ is the radius.
→ The general form of equation of circle is x2 + y2 + 2gx + 2fy + c = 0 where (-g, -f) is the centre and \(\sqrt{g^2+f^2-c}\) is the radius.
→ The intercept made by the circle x2 + y2 + 2gx + 2fy + c = 0.
- on X – axis is \(2 \sqrt{g^2-c}\) if g2 ≥ c.
- on Y – axis is \(2 \sqrt{f^2-c}\) if f2 ≥ c.
→ Equation of circle with A(x1, y1) and B(x2, y2) as the extremities of its diameter is (x – x1)(x – x2) + (y – y1)(y – y2) = 0.
→ The equation of circle passing through the three non – collinear points (x1. y2). (x2, y2) and (x3, y3) is
where c1 = – (x12 + y12), c2 = – (x22 + y22), c3 = (x32 + y32).
→ The centre of the above circle is
→ The parametric equations of circle with (h, k) as centre and radius (r ≥ 0) are given by x = h + r cos θ, y = k + r sin θ. 0 ≤ θ ≤ 2π.
→ Denoting S11 = x12 + y12 + 2gx1 + 2fy1 + c, a point P(x1, y1) is said to be an interior or an exterior or point on the circumference of the circle S = 0 according to as S11 ^ 0, conversely the result is true.
→ The power of the point P(x1, y1) w.r.t S = 0 is S11.
→ If a straight line through a point P(x1, y1) meets the circle S = 0 at A and B then the power of the point P is equal to PA. PB.
→ The length of the tangent from P(x1, y1) to a circle S = 0 is \(\sqrt{S_{11}}\).
→ If l is the perpendicular distance from the centre of the circle to the line L = 0 and ‘r’ is the radius of the circle, then the line L = 0 is said to intersects, touches or does not intersect S = 0 according to l < r, l = r or l > r.
→ The line y = mx ± r\(\sqrt{1+m^2}\) is a tangent to the circle x2 + y2 = r2 for every real value of ‘m’.
→ If P(x1, y1) and Q(x2, y2) are two points on the circle S = x2 + y2 + 2gx + 2fy + c = 0 then the equation of the secant PQ is
(xx1 + yy1 + g(x + x1) + f(y + y1) + c) + (xx2 + yy2 + g(x + x2) + f(y + y2) + c)
= x1x2 + y1y2 + g(x1 + x2) + f(y1 + y2) + c
⇒ S1 + S2 = S12
→ The equation of tangent at (x1, y1) on S = 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 ⇒ S1 = 0
→ Equation of chord joining points A ( -g + r cos θ1, – f + r sin θ1) and B ( -g + r cos θ2, – f + r sin θ2) is (x + g) cos \(\left(\frac{\theta_1+\theta_2}{2}\right)\) + (y + f)sin\(\left(\frac{\theta_1+\theta_2}{2}\right)\) = r cos \(\left(\frac{\theta_1-\theta_2}{2}\right)\)
→ Equation of the tangent at 0 of the circle S = 0 is (x + g) cos θ + (y + f) sin θ = r.
→ The equation of normal at (x1, y1) of the circle S = 0 is (x – x1) (y1 + f) – (y – y1) (x1 + g) = 0.
→ The equation of chord of contact of a point P(x,, y,) w.r.t a circle S = 0 is S1 = 0.
→ The equation of polar of a point P(x1, y1) w.r.t S = 0 is S1 = 0.
- If P(x1, y1) lies on the interior of circle polar of P is S1 = 0.
- If P(x1, y1) lies on the circle then the polar of P is the tangent given by S1 =0.
- If P(x1, y1) lies on the exterior of circle then the polar of P is the chord of contact of P is given by S1 = 0.
→ The pole of lx + my + n = 0 w.r.t S = 0 is (-g + \(\frac{{lr}^2}{\lg +\mathrm{mf}-\mathrm{n}}\), -f + \(\frac{m r^2}{l g+m f-n}\)) where r is the radius of the circle.
→ The pole of the line lx + my + n = 0 w.r.t the circle x2 + y2 = a2 is
→ Two points P and Q are said to be conjugate points w.r.t S = 0 if Q lies on the polar of P and P lies on the polar of Q.
→ The two lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate w.r.t the circle S = 0 if and only if a2(l1l2 + m2m2) = n1n2.
→ The two lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are said to be conjugate w.r.t. S = 0 if and only if r2 (l1l2 + m1m2) = (l1g + m1f – n1) (l2g + m2f – n2) where r is the radius of S = 0.
→ Two points P and Q are said to be inverse points w.r.t S = 0 if CP. CQ = r2, where C is the centre and r is the radius of the circle S = 0.
→ The equation of chord having (x,. y,) as its midpoint w.r.t circle S = 0 is S1 = S11
⇒ xx1 + yy1 + g(x + x1) + f(y + y1) + c = x12 + y12 + 2gx1 + 2fy1 + c
→ If C1, C2 are the centres of two circles S = 0, S’ = 0 and r1, r2 are the radii of two circles then
- If C1C2 > r1 + r2 we get 4 common tangents and a non – intersecting, non-touching system.
- If C1C2 = r1 + r2 we get 3 tangents where the two circles touch externally.
- If |r1 – r2| < C1C2 < r1 + r2 we get 2 tangents (intersecting system).
- If C1C2 = |r1 – r2| we get one tangent and internal touching system.
- If C1C2 < |r1 – r2| we get no tangent.
→ The combined equation of the pair of tangents drawn from an external point P(x1, y1) to the circles = 0 is SS11 = S12.
⇒ (x2 + y2 + 2gx + 2fy + c) (x12 + y12 + 2gx1 + 2fy1 + c) = [xx1 + yy1 + g(x + x1) + f(y + y1) + c]2
→ Chord of coatact: If Pis a point outside of the circle and the tangents from P touch the circle in A and B, then the chord joining the points A and B (points of contact) is called as chord of contact of P, with respect to that circle.