Circles Flashcards
Diametric form
(x-x1)(x-x2) + (y-y1)(y-y2) = 0
r =
√g^2 + f^2 - C
Power of a pt
S1 = x1^2 + y1^2 + 2gx1 + 2fy1 + C
If S1 > 0
pt lies out
Length of tangent
√S1
Intercept cut off by circle from x axis
2√g^2-C^2
Eqn of line
lx + my + n = 0
If r1+r2 < C1C2
circles don’t touch
If r1+r2>C1C2
Circles touch externally
If r1 + r2 = C1C2
Circles intersect at 2 pts
If r1-r2 = C1C2
Circles touch internally
If r1-r2 < C1C2
One circle in other
Common chord eqn
S1 - S2 = 0
director circle
x^2+y^2 = 2r^2
angle b/n circles cos(180-θ) =
[r1^2 + r2^2 - (C1C2)^2]/2r1r2
If 2 circles cut orthogonally
r1^2 + r2^2 = (C1C2)^2
No of common tangents
4 if they don’t touch
3 if they touch
2 if they intersect at 2 pts
1 if one is in other & touches
0 if one in other
Eqn of tangent to circle
xx1 + yy1 = r^2
Condition for line to be tangent to x^2 + y^2 = r^2
C^2 = r^2(1+m^2)
Parametric eqns
x = rcosθ
y = rsinθ
If e = 0
circle
If e = 1
parabola
if e < 1
ellipse
For y^2 = 4ax
focus(a,0)
directrix x = -a
LR = 4a
focal dist = |x+a|
Parametric form for y^2 = 4ax
x = at^2, y = 2at
For x^2 = 4ay
Focus(0,a)
directrix y = -a
LR = 4a
Focal dist |y+a|
Parametric form for x^2=4ay
x = 2at, y = at^2
Shift of vertex
(x-h)^2 = 4a(y-k)
Point of contact for y^2=4ax
x = a/m^2 , y = 2a/m
Point of contact for x^2=4ay
x = 2am, y = am^2
Eqn of tangent at (x1,y1) to y^2=4ax
yy1 = 2a(x+x1)
Eqn of normal at (x1,y1) to y^2=4ax
y-y1 = -y1(x+x1)/2a
For ellipse
Foci(a,0)
e = c/a
x = +-a/e
LR = 2b^2/a
c = √a^2-b^2
For ellipse, eqn of tangent
y = mx +- √a^2m^2+b^2 or mx +- √b^2m^2+a^2
Pt of contact for ellipse
-a^2m/c , b^2/c
auxillary circle
x^2 + y^2 = a^2
director circle for ellipse
x^2 + y^2 = a^2 + b^2
For hyperbola, eqn of tangent
C^2 = a^2m^2 - b^2
For hyperbola, pt of contact
+-a^2m/c, b^2/c
Director circle for hyperbola
x^2+y^2 = a^2 - b^2
In rectangular hyperbola, e =
√2
