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Circle

⚫Introduction
⚪ A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always constant. A fixed point is called the centre of the circle and the distance is called the radius of the circle.
General equation of conic : ax² + by² + 2hxy + 2gx + 2fy + c = 0
Condition for being Circle :
i) h = 0
ii) abc + 2fgh -af² - bg² -ch² ≠ 0
iii) a= b = 1
⚪ Equation of circle : x² + y² + 2gx + 2fy + c = 0
center of circle : (-g , -f)
Radius = (g² + f² -c)^1/2
If g² + f² -c > 0 , then circle will be real circle
If g² + f² -c = 0 , then circle will be point circle
If g² + f² -c < 0 , then circle will be imaginary circle

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⚫ Different forms of equations of Circle
⚪ Central form
Let C be the centre of the circle and its coordinates ( h, k ) . Let the radius of the circle be a. Let (x,y) be any point on the circumference. Then Equation of circle :
( x - h )² + ( y - k )² = a²
⚪ Standard Form : The Equation of the circle with centre at (0 , 0) and radius a is
x² + y² = a²
⚪ Circles passes through the origin :
x² + y² -2hx -2ky = 0
where a² = h² + k²
⚪ Circles touches x-axis :
x² + y² -2hx -2ay + h² = 0
where a = k
⚪ Circles touches y-axis :
x² + y² -2ax -2yk + k² = 0
where a = h
⚪ Circles touches both axes :
x² + y² -2ax -2ay + a² = 0
where a = h = k

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⚫ Equation of a circle when the co-ordinates of end points of a diameter are given where (x₁ , y₁) and (x₂ , y₂) are the points of diameter : (x - x₁ )(x - x₂) + (y - y₁ )( y - y₂) = 0

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⚫ Three point form : There is one and only one circle through three points (not on same line). If The points are (x₁, y₁), (x₂, y₂) , (x₃ ,y₃ ) , then the equation of the circle is :

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⚫ Position of point with respect to circle :

α² + β² - a² > 0 → outside
α² + β² - a² = 0 → on the circle
α² + β² - a² < 0 → inside the circle

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⚫ Position of line with respect to circle :

If P = r , line is tangent
If P < r , line cuts the circle at two points
If P > r , line doesn't cut the circle at any point

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⚫ General Equation of Tangent

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⚫ Number of tangents to a point to circle
If the point outside the circle , number of tangent is 2

If point is on the circle , number of tangents is 1

If point is inside the circle , Number of tangents is 0 (zero)

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⚫ Position of two circle with respect to each other or Number of common tangents

Common tangent = 3
(Distance Between two center)AB = r1 + r2

Distance Between two center (AB) = r1 -r2
Common tangent = 1

Distance Between two center(AB) > r1 + r2
Common Tangent = 4

Common Tangent = 2
|r1 - r2| < AB < r1 + r2

Practice set 1 Practice set 2 Practice set 3