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Revision Notes on Circle
The equation of a circle with its centre at C(x_{0}, y_{0}) and radius r is: (x – x_{0})^{2} + (y – y_{0})^{2} = r^{2}.
If x_{0} = y_{0} = 0 (i.e. the centre of the circle is at origin) then the equation of the circle reduces to x^{2} + y^{2} = r^{2}.
If r = 0 then the circle represents a point or a point circle.
The equation x^{2} + y^{2} + 2gx + 2fy + c = 0 is the general equation of a circle with centre (–g, –f) and radius √(g^{2 }+ f^{2 }- c).
Equation of the circle with points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) as extremities of a diameter is (x – x_{1}) (x – x_{2}) + (y – y_{1})(y – y_{2}) = 0.
For general circle, the equation of the chord is x_{1}x + y_{1}y + g(x_{1} + x) + f(y_{1 }+ y) + c = 0
For circle x^{2} + y^{2} = a^{2}, the equation of the chord is x_{1}x + y_{1}y = a^{2}.
The equation of the chord AB in parametric coordinates where (A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)) of the circle x^{2} + y^{2} = R^{2} is given by x cos ((α + β)/2) + y sin ((α - β)/2) = a cos ((α - β)/2).
The equation of the chord of the circle x^{2} + y^{2} = a^{2}, bisected at the point (x_{1},y_{1}) is xx_{1} + yy_{1} - a^{2} = x_{1}^{2} + y_{1}^{2} – a^{2}.
Chords are equidistant from the center of a circle if and only if they are equal in length.
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The angle subtended by an arc at the center is double the angle subtended by the same arc at the circumference of the circle.
Angle between the tangent and the radius is 90°.
Angles in the same segment are equal.
Angle in a semi-circle is 90°.
Two angles at the circumference subtended by the same arc are equal.
The below table describes the equations of circle according to changes in radii and centers:
The point P(x_{1}, y_{1}) lies outside, on, or inside a circle :S ≡ x^{2} + y^{2} + 2gx + 2fy + c = 0, according as S_{1} ≡ x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c > = or < 0.
The equation of the chord of the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 with M(x_{1}, y_{1}) as the midpoint of the chord is given by: xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) = x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} i.e. T = S_{1}.
In case the radius and the central angle of a triangle are given, the length of the chord can be computed using the formula:
The equations of tangents to the circle x^{2} + y^{2} = a^{2} in various forms:
Point Form:
If P(x_{1}, y_{1}) be any point on the circle then the equation of the tangent is given by xx_{1 }+ yy_{1} = a^{2}.
Slope form:
If the slope of the tangent is m, then the equation of the tangent is given by y = mx ± a√(1 + m^{2}).
Parametric Form:
The point P(x_{1}, y_{1}) as discussed in the point form, in parametric coordinates becomes x_{1} = a cos θ and y_{1} = a sin θ. The equation of the tangent in this case becomes x cos θ + y sin θ = a.
The equations of normal to the circle x^{2} + y^{2} = a^{2} in various forms:
If P(x_{1}, y_{1}) be any point on the circle then the equation of the tangent is given by x/x_{1 }= y/y_{1}.
The point P(x_{1}, y_{1}) as discussed in the point form, in parametric coordinates becomes x_{1} = a cos θ and y_{1} = a sin θ. The equation of the normal in this case becomes y = x tan θ.
A normal always passes through the centre of the circle.
If a circle has two secants QR and ST, then
If there is a circle which has one tangent and one secant, then the square of the tangent is equal to the product of the secant segment and its external segment.
If a radius or the diameter of a circle is perpendicular to a chord, then it divides the chord into two equal parts. The converse also holds true. Hence, in the below figure, if OB is perpendicular to PQ, then then PA = AQ.
The perpendicular bisector of a chord passes through the center of a circle. In the figure given below, OA is the perpendicular bisector of chord PQ and it passes through the circle. Similarly, OB is the perpendicular bisector of RS and it also passes through center.
In the above circle, OA is the perpendicular bisector of the chord PQ and it passes through the center of the circle. OB is the perpendicular bisector of the chord RS and it passes through the center of the circle.
If two chords in a circle are congruent, then they are equidistant from the center.
If two chords are equidistant from the center, then they are congruent. So, in the figure if PQ = RS then OA = OB and similarly, if OA = OB then PQ = RS.
The arcs intercepted by two congruent chords in a circle are congruent.
If two arcs are congruent then their corresponding chords are congruent.
If a line y = mx + c intersects the circle x^{2} + y^{2} = a^{2} in two distinct points A and B then length of intercept AB = 2√[(a^{2}(1 + m^{2}) – c^{2})/ (1 + m^{2})]
The equation of the tangent at a point on the circle and the equations of chord of contact are both represented by T = 0. But the difference between the two is that in case of chord of contact, the point say (x_{1}, y_{1}) lies outside the circle while in case of tangent it lies on the circle.
Common tangents may be either direct or transverse.
Direct common tangents to two circles meet on the line joining the centers and hence divide it externally in the ratio of their radii.
Transverse common tangents also meet on the line of centers as the direct common tangents but the difference being that they divide it internally in the ratio of their radii.
There cannot be any common tangent if a circle lies completely inside another without touching it.
If two circles touch each other internally one and only one common tangent is possible.
If two circles touch each other in two distinct points, then in such a case, two common tangents can be drawn.
In case of two circles touching each other externally, three common tangents can be drawn.
Four common tangents are possible when two circles neither touch nor intersect but one lies completely inside the other.
How to calculate direct common tangents:
Suppose there are two circles with centers at C_{1} and C_{2} and radii as r_{1}and r_{2} and P is the point of intersection of direct common tangents as shown in the figure below:
C_{1}A_{1}, C_{2}A_{2} are perpendiculars from C_{1} and C_{2} to one of the tangents. The point ‘P’ divides C_{1}C_{2} externally in the ratio r_{1}: r_{2}. So, in order to find the direct common tangent we just need to find the point P which divides the line joining the center externally in the ratio of radii.
Equation of direct common tangent is SS_{1} = T_{2} where S is the equation of one circle.
Length of C_{1}C_{2} > |r_{1} – r_{2}|
For two circles with equations S and S’, where S ≡ x^{2} + y^{2} + 2gx + 2fy + c and S’ ≡ x^{2} + y^{2} + 2g’x + 2fy’ + c’ Then equation of radical axis of two circles S = 0 and S’ = 0 is given by S = S’
2(g – g’)x + 2(f – f’)y + (c – c’) = 0
1. Internally: If |C_{1} C_{2}| = |r_{2} – r_{1}| and the point of contact is ((r_{1}x_{2 }- r_{2}x_{1})/(r_{1} + r_{2}) , (r_{1}y_{2} – r_{2}y_{1})/(r_{1} + r_{2})).
2. Externally: If |C_{1} C_{2}| = |r_{2} – r_{1}| and the point of contact is ((r_{1}x_{2} + r_{2}x_{1})/(r_{1} + r_{2}) , (r_{1}y_{2} + r_{2}y_{1})/(r_{1} + r_{2})).
For the circle S = 0, the equation, S – S’ = 0 represents the equation of the radical axis i.e. the equation of radical axis is 2x(g – g’) + 2y(f – f’) + c – c’ = 0
If the circles S = 0 and S’ = 0 intersect in real and distinct points, then S – S’ = 0 is the equation of the common chord of the two circles. If S’ = 0 and S = 0 touch each other, then S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.
The radical axis of three circles taken two at a time are concurrent and the point of concurrency is known as the radical centre.
The radical axis of two circles bisects their direct common tangents.
If two circles cut a third circle orthogonally, then the radical axis of the two circles will pass through the centre of the third circle.
If in a system of circles, every two of them have the same radical axis, then the system is a coaxial system
1.S_{1} + λ S_{2} = 0 is the family of circles passing through the intersection of S_{1} = 0 and S_{2} = 0.
2.The family of circles passing through the intersection of line L and circle S is given by S + λL = 0. A particular value of λ in the equation yields a unique circle.
3. Family of circles passing through two given points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) are given by the equation
Two circles S = 0 and S_{1} = 0 with centres at (–g, –f) and (–g_{1}, –f_{1}) cut each other orthogonally if the following equation is satisfied: 2gg_{1}+ 2ff_{1} = c + c_{1}.
Equation of polar to the circle x^{2} + y^{2} = r^{2} is xx_{1} + yy_{1} = r^{2}.
Equation of polar to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0.
Two points are said to be conjugate points with respect to a given circle if each lies on the polar of the other with respect to the circle.
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