From bubbles to bangles to doughnuts to pizzas to car wheels there are so many things which are circular in shape. No wonder circles rule our world and command so much attention.
A circle is the set of all points which are at the same distance from a given point. But, since in mathematics every curve is represented by an expression/equation so is the case with circles also. Let's try to dig more into this.
Table of Contents
A circle can be defined as the locus of a point that moves in a plane such that its distance from a fixed point in that plane is always constant.
The fixed point is known as the center of the circle and the constant distance is known as the radius of the circle.
The equation of the circle whose center is and radius ‘ is
The equation of a circle whose end-points of a diameter are and is i.e
Observe that the circle’s equation in diametric form is a combination of quadratic in having roots and quadratic in having roots .
Example : The circle passing through the point and touching the axis at also passes through the point
Example : What is the image of the circle in the line mirror ?
Step 2 :
Let be the image of w.r.t line
Where are the coefficients of and constant respectively and is the point whose image is to be found out.
Equation of the required circle is
Hence, option (a) is the correct answer.
Example : If the abscissa and ordinates of two points and are the roots of the equation
and , respectively, then find the equation of the circle with as the diameter.
Step 1 :
Let be roots of and , respectively.
Step 2 :
Equation of a circle with as the end points of the diameter is
Therefore, the equation of the required circle is
Example : What is the equation of a circle that is passing through and and having the minimum possible radius?
Step 1 :Let the equation of circle be
Since the circle passes through and , these points will satisfy its equation.
Solving and we get
The equation of circle becomes
Now the radius of the circle,
Step 2 : If is minimum then is minimum
Also, the double derivative i.e. which is positive. Hence, is a point of minima
Hence the equation of the circle is
Hence, option (d) is the correct answer.
Question1. Is a circle a two-dimensional figure?
Answer. Yes, a circle is a two dimensional shape.
Question2. If are two fixed points in a plane, then the locus of a point such that
is a straight line. Is the statement TRUE or FALSE?
Answer. False. The locus will be a circle for given conditions and will represent a straight line if
Question3. What is the circumference of a circle?
Answer. If we open a circle to form a straight line, then the length of the obtained line is the
circumference of the circle which is given by here is the radius of a circle.
Question4. What are congruent circles?
Answer. Two circles with the same radii are called congruent circles
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