01
CIRCLES
Definition of Circle:
A circle is the set of all points in a plane that are equidistant from a fixed point.
The word “Circle” comes from the Greek word
“Kirkos ” which means ring.
[Figure: Diagram of a circle with center C(h, k) and point P(x, y) on the circle]
∆ Topic-02: Equation of a Circle
In the figure, CP = r
→ √[(x − h)² + (y − k)²] = r
→ (x-h)² + ( h-k)² = r² [Squaring both sides]
→ x² + y² - 2hx - 2ky + h² + k² - r² = 0
→ x² + y² + 2gx + 2fy + c = 0
📝Here ,CP = √{ (x-h)²+(y-k)² }, This formula comes from the concept of Straight lines, where
the distance between two specific point is: √ {( second point’s abscissa - first point’s
abscissa)² + ( second point’s ordinate - first point's ordinate )² }
We get 2 formulas of a circle.
1. ( x-h )² + ( y-k )² = r²
2. x² + y² + 2gx + 2fy + c = 0 (this is the standard
equation)
Here, c = constant
-h = g , - k = f
Therefore, (h,k)→(-g,-f)
Note When the radius ( r ) is given, using the first equation makes it easier to solve.
, Condition of standard form of equation :
● Second-degree equation involving x and y
● The coefficients of x² and y² are equal to 1
● The equation has no xy term
Some example -
Correct Incorrect
x²+y²+2gx+2fy+c=0 x³+y⁴+2gx+2fy+c=0
x²+y²+2gx+2fy+c=0 4x²+5y²+2gx+2fy+c=0
x²+y²+2gx+2fy+c=0 x²+y²+2xy+2gx+2fy+c=0
∆ Topic-02 : Condition for a second degree equation to represent a circle -
ax²+by²+2hxy+Mx+ny+c = 0.
[This is a second-degree equation]
Condition
1. a=b
2. a≠0 , b≠0 [ a=b=1 ]
3. h=0
Example : 2x²+2y²+8x+6y-4=0 [here,a=b=2,h=0]
✅
or, 2(x²+y²+4x+3y-2)=0
or, x²+y²+4x+3y-2=0
↓
(standard shape)
∆ Topic -03 : Finding the center and radius from the equation of a circle -
Equation (x-h)²+(y-k)²=r² x²+y²+2gx+2gx+2fy+c=0
Center ( h,k ) ( - g,- f )
Radius r r
📝Tip: Before using these formulas, the equation must be converted to the standard form of
a circle
Example: (x – 3)² + (y + 2)² = 16
→ Center = (3, –2), Radius = 4
[This is a standard form]
Formula of Radius, r = √ [ g²+f²-c ]
📌 Remember: The radius is always positive. If r² = 25, then r = 5.
CIRCLES
Definition of Circle:
A circle is the set of all points in a plane that are equidistant from a fixed point.
The word “Circle” comes from the Greek word
“Kirkos ” which means ring.
[Figure: Diagram of a circle with center C(h, k) and point P(x, y) on the circle]
∆ Topic-02: Equation of a Circle
In the figure, CP = r
→ √[(x − h)² + (y − k)²] = r
→ (x-h)² + ( h-k)² = r² [Squaring both sides]
→ x² + y² - 2hx - 2ky + h² + k² - r² = 0
→ x² + y² + 2gx + 2fy + c = 0
📝Here ,CP = √{ (x-h)²+(y-k)² }, This formula comes from the concept of Straight lines, where
the distance between two specific point is: √ {( second point’s abscissa - first point’s
abscissa)² + ( second point’s ordinate - first point's ordinate )² }
We get 2 formulas of a circle.
1. ( x-h )² + ( y-k )² = r²
2. x² + y² + 2gx + 2fy + c = 0 (this is the standard
equation)
Here, c = constant
-h = g , - k = f
Therefore, (h,k)→(-g,-f)
Note When the radius ( r ) is given, using the first equation makes it easier to solve.
, Condition of standard form of equation :
● Second-degree equation involving x and y
● The coefficients of x² and y² are equal to 1
● The equation has no xy term
Some example -
Correct Incorrect
x²+y²+2gx+2fy+c=0 x³+y⁴+2gx+2fy+c=0
x²+y²+2gx+2fy+c=0 4x²+5y²+2gx+2fy+c=0
x²+y²+2gx+2fy+c=0 x²+y²+2xy+2gx+2fy+c=0
∆ Topic-02 : Condition for a second degree equation to represent a circle -
ax²+by²+2hxy+Mx+ny+c = 0.
[This is a second-degree equation]
Condition
1. a=b
2. a≠0 , b≠0 [ a=b=1 ]
3. h=0
Example : 2x²+2y²+8x+6y-4=0 [here,a=b=2,h=0]
✅
or, 2(x²+y²+4x+3y-2)=0
or, x²+y²+4x+3y-2=0
↓
(standard shape)
∆ Topic -03 : Finding the center and radius from the equation of a circle -
Equation (x-h)²+(y-k)²=r² x²+y²+2gx+2gx+2fy+c=0
Center ( h,k ) ( - g,- f )
Radius r r
📝Tip: Before using these formulas, the equation must be converted to the standard form of
a circle
Example: (x – 3)² + (y + 2)² = 16
→ Center = (3, –2), Radius = 4
[This is a standard form]
Formula of Radius, r = √ [ g²+f²-c ]
📌 Remember: The radius is always positive. If r² = 25, then r = 5.
