Circles and Systems of Circles: IITJEE mains co-ordinate geometry basics

Section I:

Definition of a Circle:

A circle is the locus of  a point which moves in a plane so that it’s distance from a fixed point in the plane is always constant.The fixed point is called the centre of the circle and the constant distance is called its radius.

Section II:

Equations of a circle:

  • An equation of a circle with centre (h,k) and radius r is (x-h)^{2}+(y-k)^{2}=r^{2}.
  • An equation of a circle with centre (0,0) and radius r is x^{2}+y^{2}=r^{2}.
  • An equation of a circle on the line segment joining (x_{1},y_{1}) and (x_{2},y_{2}) as diameter is (x-x_{1})(x-x_{2})+(y-y_{1})(y-y_{2})=0.
  • General equation of a circle is :x^{2}+y^{2}+2gx+2fy+c=0 where g, f, and c are constants
    • centre of this circle is (-g,-f)
    • Its radius is \sqrt{g^{2}+f^{2}-c}, g^{2}+f^{2}\geq c
    • Length of the intercept made by this circle on the x-axis is 2\sqrt{g^{2}-c} if g^{2}-c \geq 0 and that on the y-axis is \sqrt{f^{2}-c} if f^{2}-c \geq 0.
  • General equation of second order degree ax^{2}+2hxy+by^{2}+2gx+2fy+c=0 in x and y represents a circle if and only if:
    • coefficient of x^{2} equals coefficient of y^{2}, that is, a=b \neq 0
    • coefficient of xy is zero, that is , h=0
    • g^{2}+f^{2}-ac \geq 0

Section III: Some results regarding circles:

  • Position of a point with respect to a circle: Point P(x_{1},y_{1}) lies outside, on or inside a circle S \equiv x^{2}+y^{2}+2gx+2fy+c=0, according as S_{1} \equiv x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c>, =, \hspace{0.1in} or \hspace{0.1in}< 0
  • Parametric coordinates of any point on the circle (x-h)^{2}+(y-k)^{2}=r^{2} are given by (h+r\cos{\theta},k+r\sin{\theta}) with 0 \leq \theta < 2\pi. In particular, parametric coordinates of any point on the circle.
  • An equation of the tangent to the circle x^{2}+y^{2}+2gx+2fy+c=0 at the point (x_{1},y_{1}) on the circle is xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c=0
  • An equation of the normal to the circle x^{2}+y^{2}+2gx+2fy+c=0 at the point (x_{1},y_{1}) on the circle is \frac{y-y_{1}}{y_{1}+f} = \frac{x-x_{1}}{x_{1}+g}
  • Equations of the tangent and normal to the circle x^{2}+y^{2}=r^{2} at the point (x_{1},y_{1}) on the circle are, respectively, xx_{1}+yy_{1}=r^{2} and \frac{x}{x_{1}}=\frac{y}{y_{1}}
  • The line y=mx+c is a tangent to the circle x^{2}+y^{2}=r^{2} if and only if c^{2}=r^{2}(1+m^{2}).
  • The lines y=mx \pm r\sqrt{(1+m^{2})} are tangents to the circle x^{2}+y^{2}=r^{2}, for all finite values of m. If m is infinite, the tangents are x \pm r=0.
  • An equation of the chord of the circle S=x^{2}+y^{2}+2gx+2fy+c=0, whose mid-point is (x_{1},y_{1}) is T=S_{1}, where T \equiv xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c and S_{1}=x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c. In particular, an equation of the chord of the circle x^{2}+y^{2}=r^{2}, whose mid-point is (x_{1},y_{1}) is xx_{1}+yy_{1}=x_{1}^{2}+y_{1}^{2}.
  • An equation of the chord of contact of the tangents drawn from a point (x_{1},y_{1}) outside the circle S=0 is T=0.(S and T are as defined in (8) above).
  • Length of the tangent drawn from a point (x_{1},y_{1}) outside the circle S=0, to the circle, is \sqrt{S_{1}}. (S and \sqrt{S_{1}}) are as defined in (8) above.)
  • Two circles with centres C_{1}(x_{1},y_{1}) and C_{2}(x_{2},y_{2}) and radii r_{1}, r_{2} respectively, (i) touch each other externally if C_{1}C_{2}|=r_{1}+r_{2}. the point of contact is (\frac{r_{1}x_{2}+r_{2}x_{1}}{r_{1}+r_{2}},\frac{r_{1}y_{2}+r_{2}y_{1}}{r_{1}+r_{2}}) and (ii) touch each other internally if |C_{1}C_{2}|=|r_{1}-r_{2}|, where r_{1} \neq r_{2}; the point of contact is (\frac{r_{1}x_{2}-r_{2}x_{1}}{r_{1}-r_{2}} , \frac{r_{1}y_{2}-r_{2}y_{1}}{r_{1}-r_{2}})
  • An equation of the family of circles passing through the points (x_{1},y_{1}) and (x_{2},y_{2}) is (x-x_{1})(x-x_{2}) + (y-y_{1})(y-y_{2}) +\lambda(F)=0, where

F=\left|\begin{array}{ccc}    x & y & 1 \\    x_{1} & y_{1} & 1 \\    x_{2} & y_{2} & 1 \end{array}\right|

  • An equation of the family of circles which touch the line y-y_{1}=m(x-x_{1}) at (x_{1}, y_{1}) for any finite value of m is (x-x_{1})^{2}+(y-y_{1})^{2}+\lambda ((y-y_{1})-m(x-x_{1}))=0. If m is infinite, the equation becomes (x-x_{1})^{2}+(y-y_{1})^{2}+\lambda (x-x_{1})=0.
  • Let QR be a chord of a circle passing through the point P(x_{1},y_{1}) and let the tangents at the extremities Q and R of this chord intersect at the point L(h,k). Then, locus of L is called the polar of P with respect to the circle, and P is called the pole of its polar.
    • Equation of the polar of P(x_{1},y_{1}) with respect to the circle S \equiv x^{2}+y^{2}+2gx+2fy+c=0 is T=0, where T is defined as above.
    • If the polar of P with respect to a circle passes through Q, then the polar of Q with respect to the same circle passes through P. Two such points P and Q, are called conjugate points of the same circle.
  • If lengths of the tangents drawn from a point P to the two circles S_{1} \equiv x^{2}+y^{2}+2g_{1}x+2f_{1}y+c_{1}=0 and S_{2} \equiv x^{2}+y^{2}+2g_{2}x+2f_{2}y+c_{2}=0 are equal, then the locus of P is called the radical axis of the two circles S_{1} =0 and S_{2}=0, and its equation is S_{1}-S_{2}=0, that is, 2(g_{1}-g_{2})+2(f_{1}-f_{2})y+(c_{1}-c_{2})=0
    • Radical axis of two circles is perpendicular to the line joining their circles.
    • Radical axes of three circles, taken in pairs, pass through a fixed point called the radical centre of the three circles, if the centres of these circles are non-collinear.

4: Special Forms of Equation of a Circle:

  1. An equation of a circle with centre (r,r) and radius |r| is (x-r)^{2}+(y-r)^{2}=r^{2}. This touches the co-ordinate axes at the points (r,0) and (0,r).
  2. An equation of a circle with centre (x_{1},r), radius |r| is (x-x_{1})^{2}+(y-r)^{2}=r^{2}. This touches the x-axis at (x_{1},0).
  3. An equation of a circle with centre (\frac{a}{2},\frac{b}{2}) and radius \sqrt{\frac{(a^{2}+b^{2})}{4}} is x^{2}+y^{2}-ax-by=0. This circle passes through the origin (0,0), and has intercepts a and b on the x and y axes, respectively.

5: Systems of Circles:

Let S \equiv x^{2}+y^{2}+2gx+2fy+c; and S^{'} \equiv x^{2}+y^{2}+2g^{'}x+2f^{'}y+c^{'} and L \equiv ax + by + k^{'}.

  1. If two circles S=0 and S^{'}=0 intersect at real and distinct points, then S+\lambda S^{'}=0 where \lambda \neq -1 represents a family of circles passing through these points, where \lambda is a parameter, and S-S^{'}=0 when \lambda=-1 represents the chord of the circles.
  2. If two circles S =0 and S^{'}=0 touch each other, then S-S^{'}=0 represents equation of the common tangent to the two circles at their point of contact.
  3. If two circles S=0 and S^{'}=0 intersect each other orthogonally (the tangents at the point of intersection of the two circles are at right angles), then 2gg^{'}+2ff^{'}=c+c^{'}.
  4. If the circle S=0 intersects the line L=0 at two real and distinct points, then S+\lambda L=0 represents a family of circles passing through these points.
  5. If L=0 is a tangent to the circle S=0 at P, then S+\lambda L=0 represents a family of circles touching S=0 at P, and having L=0 as the common tangent at P.
  6. Coaxial Circles: A system of circles is said to be coaxial if every pair of circles of the system have the same radical axis. The simplest form of the equation of a coaxial system of circles is : x^{2}+y^{2}+2gx+c=0, where g is a variable and c is constant, the common radical axis of the system being y-axis and the line of centres being x-axis.  The Limiting points of the coaxial system of circles are the members of the system which are of zero radius. Thus, the limiting points of the coaxial system of circles x^{2}+y^{2}+2gx+c=0 are (\pm \sqrt{c},0) if c>0. The equation S+\lambda S^{'}=0 (\lambda \neq -1) represents a family of coaxial circles, two of whose members are given to be S=0 and S^{'}=0.
  7. Conjugate systems (or orthogonal systems) of circles : Two system of circles such that every circle of one system cuts every circle of  the other system orthogonally are said to  be conjugate system of circles. For instance, x^{2}+y^{2}+2gx+c=0 and x^{2}+y^{2}+2fy-c=0, where g and f are variables and c is constant, represent two systems of coaxial circles which are conjugate.

6: Common tangents to two circles:

If (x-g_{1})^{2}+(y-f_{1})^{2}=a_{1}^{2} and (x-g_{2})^{2}+(y-f_{2})^{2}=a_{2}^{2} are two circles with centres C_{1}(g_{1},f_{1}) and C_{2}(g_{2},f_{2}) and radii a_{1} and a_{2} respectively, then we have the following results regarding their common tangents:

  1. When C_{1}C_{2}>a_{1}+a_{2}, that is, distance between the centres is greater than the sum of their radii, the two circles do not intersect with each other, and four common tangents can be drawn to circles. Two of them are direct common tangents and other two are transverse common tangents. The points T_{1},T_{2} of intersection of direct common tangents and transverse common tangents respectively, always lie on the line joining the centres of the two circles and divide it externally and internally respectively in the ratio of their radii.
  2. When C_{1}C_{2}=a_{1}+a_{2}, that is, the distance between the centres is equal to the sum of their radii, the two circles touch each other externally, two direct tangents are real and distinct and the transverse tangents coincide.
  3. When C_{1}C_{2}<a_{1}+a_{2}, that is, the distance between the centres is less than the sum of the radii, the circles intersect at two real and distinct points, the two direct common tangents are real and distinct while the transverse common tangents are imaginary.
  4. When C_{1}C_{2}=|a_{1}-a_{2}| with a_{1} \neq a_{2}, that is, the distance between the centres is equal to the difference of their radii, the circles touch each other internally, two direct common tangents are real and coincident, while the transeverse common tangents are imaginary.
  5. When C_{1}C_{2}<|a_{1}-a_{2}|, with a_{1} \neq a_{2}, that is, the distance between the centres is less than the difference of the radii, one circle with smaller radius lies inside the other and the four common tangents are all imaginary.

To be continued,

Nalin Pithwa.

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