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
2.3 Parametric 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
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
If the radius of a circle whose centre $O$ is at $(0,0)$ makes an angle $\theta $ with the positive direction of $X$-axis, then $\theta $ is called the parameter.
Let $OP=a$, then $OM=a\cos \theta$ and $PM = a\sin \theta $
i.e., $$x = a\cos \theta {\text{, }}y = a\sin \theta $$
Hence, $(x,y) \to (a\cos \theta ,a\sin \theta )$ are the parametric coordinates of the circle $${x^2} + {y^2} = {a^2},{\text{ }}0 \leqslant \theta \leqslant 2\pi $$
If the centre of circle is $(h,k)$, then parametric coordinates of any point $(x,y)$ on the circle is given by $$x = h + a\cos \theta ,{\text{ }}y = k + a\sin \theta $$
where $0 \leqslant \theta \leqslant 2\pi $ and the parametric equation of circle is $${(x - h)^2} + {(y - k)^2} = {a^2}$$
Question 1. Find the parametric form of the equation of the circle ${x^2} + {y^2} + px + py = 0$.
Solution: Equation of the circle can be re-written in the form $${\left( {x + \frac{p}{2}} \right)^2} + {\left( {y + \frac{p}{2}} \right)^2} = \frac{{{p^2}}}{2}$$
From the above central form of circle, centre of circle is $\left( { - \frac{p}{2}, - \frac{p}{2}} \right)$
Therefore, parametric coordinates are $$x = - \frac{p}{2} + \frac{p}{{\surd 2}}cos\theta = \frac{p}{2}\left( { - 1 + \sqrt 2 cos\theta } \right)$$ and $$y = \frac{{ - p}}{2} + \frac{p}{{\sqrt 2 }}\cos \theta = \frac{p}{2}( - 1 + \sqrt 2 \sin \theta )$$ where $0 \leqslant \theta \leqslant 2\pi $.
Question 2. If the parametric form of a circle is given by $x = - 4 + 5\cos \theta $ and $y = - 3 + 5\sin \theta $. Find the cartesian form of circle.
Solution: The given equations are $x = - 4 + 5\cos \theta $ and $y = - 3 + 5\sin \theta $
or, $x + 4 = 5\cos \theta...(1) {\text{ and }}y + 3 = 5\sin \theta...(2) $
Squaring and adding equations $(1)$ and $(2)$, we get $${\left( {x + 4} \right)^2} + {\left( {y + 3} \right)^2} = {5^2}$$ or, $${\left( {x + 4} \right)^2} + {\left( {y + 3} \right)^2} = 25$$
Question 3. The equation of circle is ${x^2} + {y^2} = 16$ and $P$ is a variable point on the circle. Coordinates of another point $A$ which a lie outside the circle is $(5,1)$. Find the locus of midpoint of a line joining $P$ and $A$ i.e., $PA$.
Solution: Let us assume the parametric coordinates of point $P$ on a circle be $\left( {4cos\theta ,4sin\theta } \right)$. Coordinates of mid-point of $PA$ be $M(h,k)$.
Using mid-point formulae, we can write $$h = \frac{{4cos\theta + 5}}{2}\ \ and\ \ k = \frac{{4sin\theta + 1}}{2}$$
or, $$4cos\theta = 2h - 5...(1)$$ $$4sin\theta = 2k - 1...(2)$$
Squaring and adding $(1)$ and $(2)$, we get $${\left( {2h - 5} \right)^2} + {\left( {2k - 1} \right)^2} = 16\left( {co{s^2}\theta + si{n^2}\theta } \right)$$ $${\left( {2h - 5} \right)^2} + {\left( {2k - 1} \right)^2} = 16$$ $${\left( {h - \frac{5}{2}} \right)^2} + {\left( {k - \frac{1}{2}} \right)^2} = {2^2}$$
which is the equation of circle. So, we can conclude that the locus of midpoint of a line joining $P$ and $A$ is a circle with centre $(\frac{5}{2},\frac{1}{2})$ and radius $2$ units.
