Circles

Question 1

General eqn of a circle

Answer 1

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

Question 2

Eqn for y intercept of a circle

Answer 2

In circle x² + y² + 2gx + 2fy + c = 0

2√(f² - c)

Question 3

Eqn for x intercept of a circle

Answer 3

In x² + y² + 2gx + 2fy + c = 0

2 √(g² - c)

Question 4

Condition for circle to touch y axis

Answer 4

f² = c

Question 5

Condition for circle to touch x axis

Answer 5

g² = c

Question 6

Eqn of circle passing through (0,0) (a,0) (0,b)

Answer 6

x² + y² -ax -by = 0

Question 7

Circle having (x1, y1) and (x2, y2) as diameter is

Answer 7

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

Question 8

Parametric form of the eqn
(x-h)² + (y-k)² = r²

Answer 8

• x = h + r cosθ
• y = k + r sinθ

Question 9

Shortest distance between two circles

Answer 9

S.D= C₁C₂ - (r₁ +r₂)

• C₁ and C₂ are centres of circle
• r₁ and r₂ are radius of circle

Question 10

Max distance between two circles

Answer 10

M.D= C₁C₂ + (r₁ +r₂)

• C₁ and C₂ are centres of circle
• r₁ and r₂ are radius of circle

Question 11

Length of tangent from a point P(x₁, y₁) to a circle

Answer 11

Length=√ (x₁² + y₁² -r²)

• r: radius of the circle

Substitute value of x1 and y1 in the eqn of the circle

Question 12

Circumcentre

Answer 12

• Acute ∆le: inside
• Right ∆le: midpoint of hypotenuse
• obtuse ∆le: outside the ∆le

Question 13

Trick to find the coordinates of right angle if three points are given

Answer 13

Out of the three it will be the one having common x and y coordinates

Question 14

Identifying point lying inside or outside circle

Answer 14

• 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

Question 15

Radius of circle
x² + y² + 2gx + 2fy + c = 0

Answer 15

R = √ (g² + f² - C)

Question 16

Common tangents condition

Answer 16

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

Question 17

Equation of normal to the circle
x² + y² = r² at point (x1, y1)

Answer 17

x/x1 = y/y1

Question 18

Equation of tangent to a circle
x² + y² + 2gx + 2fy + c =0 from (x1, y1)

Answer 18

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

Question 19

Checking whether a circle is real or imaginary

Answer 19

• g² + f² - c > 0: real
• g² + f² - c = 0: reduces to a point
• g² + f² - c < 0: imaginary

Question 20

Equation of chord of contact of a circle

Answer 20

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

The chord is formed by the tangents drawn from a point P(x₁, y₁)

Question 21

Condition for orthogonality of two circles

Answer 21

• 2(g₁g₂ + f₁f₂) = c₁ + c₂

orthogonally cutting means the angle between the two circles is 90°

Question 22

Equation of pair of tangents from point P(x₁, y₁)

Answer 22

SS₁ = T²

• S: formula of circle
  [x² + y² + 2gx + 2fy + c]
• S₁: substitute value of point in eqn of circle
  [x₁² + y₁² + 2gx₁ + 2fy₁ + c]
• T: eqn of tangent
  [xx₁² + yy₁² + 2g(x+ x₁) + 2f(y+y₁) + c]