Circles Flashcards
General eqn of a circle
x² + y² + 2gx + 2fy + c = 0
- centre (-2g/2, -2f/2)
- radius √(g²+f² - c)
(x-h)² + (y-k)² = r²
- (h,k) centre
- r: radius
Eqn for y intercept of a circle
In circle x² + y² + 2gx + 2fy + c = 0
2√(f² - c)
Eqn for x intercept of a circle
In x² + y² + 2gx + 2fy + c = 0
2 √(g² - c)
Condition for circle to touch y axis
f² = c
Condition for circle to touch x axis
g² = c
Eqn of circle passing through (0,0) (a,0) (0,b)
x² + y² -ax -by = 0
Circle having (x1, y1) and (x2, y2) as diameter is
(x - x1)(x - x2) + (y - y1)(y - y2) = 0
Parametric form of the eqn
(x-h)² + (y-k)² = r²
- x = h + r cosθ
- y = k + r sinθ
Shortest distance between two circles
S.D= C₁C₂ - (r₁ +r₂)
- C₁ and C₂ are centres of circle
- r₁ and r₂ are radius of circle
Max distance between two circles
M.D= C₁C₂ + (r₁ +r₂)
- C₁ and C₂ are centres of circle
- r₁ and r₂ are radius of circle
Length of tangent from a point P(x₁, y₁)
Length=√ (x₁² + y₁² -r²)
- r: radius of the circle
Substitute value of x1 and y1 in the eqn of the circle
Circumcentre
- Acute ∆le: inside
- Right ∆le: midpoint of hypotenuse
- obtuse ∆le: outside the ∆le
Trick to find the coordinates of right angle if three points are given
Out of the three it will be the one having common x and y coordinates
Identifying point lying inside or outside circle
- C(x1, y1) ≤ 0: point inside or on the circle
- C(x1, y1) > 0: point outside the circle
Substitute x1 and y1 in the eqn of the circle
Radius of circle
x² + y² + 2gx + 2fy + c = 0
R = √ (g² + f² - C)
Common tangents condition
Disjoint circle (4 tangents)
- C1C2 > r1 + r2
Externally touching circle (3 tangent)
- C1C2 = r1 + r2
Intersecting (2 tangents)
- C1C2 < r1 + r2
Internally touching (1 tangent)
- C1C2 = r2 - r1
Equation of normal to the circle
x² + y² = r² at point (x1, y1)
x/x1 = y/y1
Equation of tangent to a circle
x² + y² + 2gx + 2fy + c =0 from (x1, y1)
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
Checking whether a circle is real or imaginary
- g² + f² - c > 0: real
- g² + f² - c = 0: reduces to a point
- g² + f² - c < 0: imaginary
