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Mathematics : Two Dimensional Analytical Geometry II: Summary

SUMMARY

(1)      Equation of the circle in a standard form is (x - h)2 + ( y - k )2 = r 2 .

(i)        Centre (h, k) (ii) radius ‘ r

(2)      Equation of a circle in general form is x2 + y2 + 2gx + 2 fy + c = 0 .

(i)        centre (-g, - f ) (ii) radius = √[g 2 + f 2 c]

(3)      The circle through the intersection of the line lx + my + n = 0 and the circle

x2 + y2 + 2gx + 2 fy + c = 0 is x2 + y2 + 2gx + 2 fy + c + λ(lx + my + n) = 0, λ  R1 .

(4)      Equation of a circle with (x1 , y1 ) and (x2 , y2 ) as extremities of one of the diameters is

(x - x1)(x - x2 ) + ( y - y1 )( y - y2 ) = 0 .

(5)      Equation of tangent at (x1 , y1 ) on circle x2 + y2 + 2gx + 2fy + c = 0 is

xx1 + yy1 + g(x + x1 ) + f ( y + y1 ) + c = 0

(6)      Equation of normal at (x1 , y1 ) on circle x2 + y2 + 2gx + 2 fy + c = 0 is

yx1 - xy1 + g( y - y1 ) - f (x - x1) = 0 .

Table 1

Tangent and normal

Table 2

Condition for the sine  y = mx + c to be a tangent to the Conics

Table 3

Parametric forms

Identifying the conic from the general equation of conic Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The graph of the second degree equation is one of a circle, parabola, an ellipse, a hyperbola, a point, an empty set, a single line or a pair of lines. When,

(1) A = C = 1, B = 0, D = -2h, E = -2k, F = h2 + k 2 - r 2  the general equation reduces to (x - h)2 + ( y - k )2 = r2 , which is a circle.

(2)      B = 0 and either A or C = 0 , the general equation yields a parabola under study, at this level.

(3)      A  ≠  C and A and C are of the same sign the general equation yields an ellipse.

(4)      A  ≠ C and A and C are of opposite signs the general equation yields a hyperbola

(5)      A = C and B = D = E = F = 0 , the general equation yields a point x2 + y2 = 0 .

(6)      A = C = F and B = D = E = 0 , the general equation yields an empty set x2 + y2 +1 = 0 , as there is no real solution.

(7)      A  ≠  0 or C 0 and others are zeros, the general equation yield coordinate axes.

(8)      A = -C and rests are zero, the general equation yields a pair of lines x2 - y2 = 0 .

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12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : Summary | Equation, Formula | Two Dimensional Analytical Geometry II