Circles
1.0 Definition
2.0 Equation of circle in various forms
2.1 Central Form
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
3.0 Intercepts made by a circle on coordinate axis
4.0 Position of a point with respect to a circle
5.0 Maximum and minimum distance of a point from a circle
6.0 Intersection of a line and a circle
7.0 Length of intercept cutoff from a line by a circle or length of chord of a circle
8.0 Equation of tangent to a circle
8.1 Slope form
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
9.0 Tangents from a point to the circle
10.0 Length of tangent from a point to a circle
11.0 Common Tangents
12.0 Equation of common tangents
13.0 Pair of tangents
14.0 Normal to a circle at a given point
15.0 Common chord of two circles
16.0 Equation of chord joining two points on circle
17.0 Equation of chord of circle whose midpoint is given
18.0 Chord of contact
19.0 Orthogonal Circles
20.0 Director Circle
21.0 Family of circles
2.4.1 Application of general form of circle
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
To find the equation of circle passing through three non-collinear points. (CONCEPT THROUGH QUESTIONS)
Method 1: Find the equation of circle passing through $(1,0)$, $(-1,0)$ and $(0,1)$.
Solution: Let the equation of circle passing through the given $3$ points in general form be $${x^2} + {y^2} + 2gx + 2fy + c = 0$$
The above equation satisfies all the $3$ points. Putting the values, we get
$${1^2} + {0^2} + 2 \times g \times 1 + 2 \times f \times 0 + c = 0$$ $$2g + c = - 1 \ldots \left( 1 \right)$$ $$ - {1^2} + {0^2} + 2 \times g \times - 1 + 2 \times f \times 0 + c = 0$$ $$ - 2g + c = - 1 \ldots \left( 2 \right)$$ $${0^2} + {1^2} + 2 \times g \times 0 + 2 \times f \times 1 + c = 0$$ $$2f + c = - 1 \ldots \left( 3 \right)$$
Adding $(1)$ and $(2)$, we get $c=-1$.
Subtracting $(1)$ and $(2)$, we get $g=0$.
Put the value of $c$ in $(3)$, we get $f=0$.
Therefore, the equation of circle passing through the given $3$ points is ${x^2} + {y^2} - 1 = 0$.
Method 2: Equation of circle passing through $(1,1)$, $(2,-1)$ and $(3,2)$.
Solution: Write the equation of circle in the form as shown below
$$\begin{gathered}\left| {\begin{array}{c}{{x^2} + {y^2}}&x&y&1 \\ {{1^2} + {1^2}}&1&1&1 \\ {{2^2} + {{\left( { - 1} \right)}^2}}&2&{ - 1}&1 \\ {{3^2} + {2^2}}&3&2&1 \end{array}} \right| = 0 \hspace{1em} \\\left| {\begin{array}{c}{{x^2} + {y^2}}&x&y&1 \\ 2&1&1&1 \\ 5&2&{ - 1}&1 \\ {13}&3&2&1\end{array}} \right| = 0 \hspace{1em} \\ \end{gathered} $$
Applying ${C_1} \to {C_1} - 2{C_4},{\text{ }}{C_2} \to {C_2} - {C_4}{\text{ and }}{C_3} \to {C_3} - {C_4}$
$$\left| {\begin{array}{c}{{x^2} + {y^2} - 2}&{x - 1}&{y - 1}&1 \\ 0&0&0&1 \\ 3&1&{ - 2}&1 \\ {11}&2&1&1 \end{array}} \right| = 0$$
Expand with respect to second row, we get
$$\left| {\begin{array}{c}{{x^2} + {y^2} - 2}&{x - 1}&{y - 1} \\ 3&1&{ - 2} \\ {11}&2&1 \end{array}} \right| = 0$$
Applying ${C_1} \to {C_1} - 3{C_2}{\text{ and }}{C_3} \to {C_3} - 2{C_2}$
$$\left| {\begin{array}{c}{{x^2} + {y^2} - 3x + 1}&{x - 1}&{2x + y - 3} \\ 0&1&0 \\ 5&2&5 \end{array}} \right| = 0$$
Expand with respect to second row, then
$$\left| {\begin{array}{c}{{x^2} + {y^2} - 3x + 1}&{2x + y - 3} \\ 5&5 \end{array}} \right| = 0$$ or, $$5\left| {\begin{array}{c}{{x^2} + {y^2} - 3x + 1}&{2x + y - 3} \\ 1&1 \end{array}} \right| = 0$$ $$\left| {\begin{array}{c}{{x^2} + {y^2} - 3x + 1}&{2x + y - 3} \\ 1&1 \end{array}} \right| = 0$$ $${x^2} + {y^2} - 3x + 1 - 2x - y + 3 = 0$$ $${x^2} + {y^2} - 5x - y + 4 = 0$$
