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
15.1 Length of common chord
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
As shown in figure $45$, $PQ=2PM$$ = 2\sqrt {{{\left( {{C_1}P} \right)}^2} - {{({C_1}M)}^2}} $
where
${C_1}P = $radius of circle$S=0$ and
${C_1}M = $length of perpendicular from ${C_1}$ on common chord $PQ$
NOTE: To find the length of common chord, first we have to find the equation of common chord if not given in the question.
Question 24. $S \equiv {x^2} + {y^2} - 6x - 4y + 9 = 0$ and $S' \equiv {x^2} + {y^2} - 8x - 6y + 23 = 0$ are two circles intersecting each other. Find the equation and length of common chord.
Solution: The equation of common chord is $S-S'=0$ i.e., $$({x^2} + {y^2} - 6x - 4y + 9) - \left( {{x^2} + {y^2} - 8x - 6y + 23} \right) = 0$$ $$2x+2y=14$$ or, $$x+y=7$$
Now, from figure $46$, Radius of circle $S=0$ i.e., ${r_2} = \sqrt {9 + 4 - 9} = 2 = {C_2}P$.
And ${C_2}M = $ perpendicular distance from centre ${C_2}$ to the common chord $PQ = \left| {\frac{{3 + 2 - 7}}{{\sqrt {1 + 1} }}} \right| = \sqrt 2 $.
In $\Delta {C_2}MP$, use Pythagoras theorem, we get $$M{P^2} = {C_2}{P^2} - {C_2}{M^2}$$ or, $$MP = \sqrt {{2^2} - {{\sqrt 2 }^2}} = \sqrt 2 $$
Therefore, length of common chord $PQ = 2MP = 2\sqrt 2 $.
Question 25. If the circle $S \equiv {x^2} + {y^2} + 4x + 22y + c = 0$ bisects the circumference of circle $S' \equiv {x^2} + {y^2} - 2x + 8y - d = 0$. Find the value of $c+d$.
Solution: If the circle bisects the circumference of other circle, the common chord passes through the centre of the bisected circle as shown in figure $47$.
The equation of common chord $PQ$ is $S-S'=0$ i.e., $$({x^2} + {y^2} + 4x + 22y + c) - \left( {{x^2} + {y^2} - 2x + 8y - d} \right) = 0$$ or, $$6x + 14y + d + c = 0$$ $$c + d = - 6x - 14y...(1)$$
The equation $(1)$ satisfies the centre ${C_2}(1, - 4)$, we get $$c + d = - 6 \times 1 - 14 \times - 4 = - 6 + 52 = 46$$
