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 *x*^{2} + *y*^{2}
+ 2*gx** *+ 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

*x*^{2} + *y*^{2} + 2*gx** *+ 2 *fy *+ *c *= 0 is *x*^{2} + *y*^{2} + 2*gx** *+ 2 *fy *+ *c *+ *λ*(*lx** *+ *my *+
*n*) = 0, *λ **∈ * **R**^{1} .

(4)
Equation of a circle with (*x*_{1} , *y*_{1}
) and (*x*_{2} , *y*_{2} ) as extremities of one of the diameters is

(*x *- *x*_{1})(*x *- *x*_{2} ) + ( *y *- *y*_{1}
)( *y *- *y*_{2} ) = 0 .

(5)
Equation of tangent at (*x*_{1}
, *y*_{1}
) on circle *x*^{2} + *y*^{2} + 2*gx *+ 2*fy *+ *c *= 0 is

*xx*_{1} + *yy*_{1} + *g*(*x *+ *x*_{1}
) + *f *( *y *+ *y*_{1} ) + *c *= 0

(6)
Equation of normal at (*x*_{1}
, *y*_{1}
) on circle *x*^{2} + *y*^{2} + 2*gx *+ 2 *fy *+ *c *= 0 is

*yx*_{1} - *xy*_{1} + *g*( *y
*- *y*_{1} ) - *f *(*x
*- *x*_{1}) = 0 .

**Table 1**

**Tangent and normal**

**Table 2**

**Condition for the sine **** y **=

**Table 3**

**Parametric forms **

**Identifying the conic from the general equation of conic Ax^{2} **+

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 *= -2*h*, *E *= -2*k*, *F *= *h*^{2} + *k *^{2}
- *r *^{2}
the general equation reduces to (*x
*- *h*)^{2} + ( *y *- *k
*)^{2} = *r*^{2} ,
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 *x*^{2} + *y*^{2} = 0 .

(6) *A *= *C *= *F *and *B *= *D *= *E *= 0 , the general equation yields an
empty set *x*^{2} + *y*^{2} +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 *x*^{2} - *y*^{2}
= 0 .

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

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

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