In this article, we will cover the concept of the parametric form of a circle. This concept falls under the broader category of coordinate geometry. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including two in 2021.
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To represent any point on a curve in terms of a single variable (parameter), we use parametric form of that curve.
1. Parametric Form for
Here as
So any arbitrary point on this circle can be assumed as
2. Parametric Form for
Centre of the circle here is
Parametric point on it is
Example 1: Let
1) a parabola
2) a straight line
3) a hyperbola
4) an ellipse
Solution:
Let P be any point on the circle
So,
is maximum if
Hence, the answer is the option 2.
Example 2: What are the parametric coordinates of
Solution:
As we have learned
Parametric form for
Since, in
Thus
Example 3: Which of the following is true for the Circles
1)
2)
3)
4) All of the above
Solution:
So they are concentric for any value of a.
Now radius for the first circle
The radius for the second circle
- For
radius of the second circle > radius of the first circle
As they are concentric, so in this case the first circle lies entirely inside the second circle
- For
radius of second circle < radius of first circle
As they are concentric, so in this case, the second circle lies entirely inside the first circle
Hence, the answer is the option 4.
Example 4: What are the parametric coordinates for
Solution:
As we have learned
Parametric form:
For a circle having centre
Grample 5: Two circles are inscribed and circumscribed about a square A B C D, the length of each side of the square is 16 .
Solution:
Let the center of the square be the origin
The radii of the inscribed and circumscribed circles are respectively 8 and
Let the coordinate of
Then
Similarly
Hence, the answer is 1792 .
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