The equation of a circle is what we already know but the equation of the same circle can be written in different ways, one of which is parametric form.
While solving problems on circles it is often required to consider a general point which lies on the circle. Here both and are variables. Instead of considering two variables, parametric form of the circle helps us convert this general point into a single variable θ. So it is very helpful and we can solve problems easily by considering the parametric form of the circle. Let’s have a look at it to understand it in a better way.
Table of Contents
Length of the intercept made by the circle
Note : General equation of the circle for different conditions:
Example : For how many values of , does the circle and the coordinate axes have exactly three common points?
Comparing with the general equation of the circle, we get
The center of the circle is
Geometrically, the circle will have exactly three common points with coordinate axes in the following cases :
Hence, only two values of is possible.
Example : Circle(s) touching the axis at a distance of units on the positive side from origin and having intercept of length on axis is/are as follows :
Options (a),(c) are correct
Example : If the point holds good for , then find the maximum and minimum values of .
Step 1 :
Given, equation of the circle,
Comparing with ,
Center, and radius
Now, the parametric coordinates of a point on the given circle,
Step 2 :
We know that,
Therefore, , and
Example : Find the circumcenter of if , and .
Vertices of are given in the form of parametric coordinates for a circle that is , comparing with any one vertex of , we get
Therefore center of circle circumcentre of triangle
Hence, Circumcenter is
Question 1.What is the radius vector of a circle?
Answer: line segment joining the center and a point on the circumference of a circle is called the radius vector of the circle.
Question 2. Does the radius of the circle change by just shifting the center of the circle?
Answer: Since the radius is the distance between the center and circumference of the circle, it remains constant.
Question 3. Can a circle make an equal intercept on both axes?
Answer: Yes, if the value of and in the general form of the circle are equal, it will make an equal intercept on both the axes.
Question 4. What is the difference between the parametric form of a straight line and the parametric form of the circle?
Answer: The parametric form of a circle is , Here ( is a parameter and is a constant whereas the parametric form of a straight line passing through is , Here is a parameter and is the angle which the straight line makes with the positive direction of
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