Understanding the topic of circles and their equation is important, as it has a significant wide range of real-life applications, such as the use of geometry while designing wheels, designing of circular tracks and many more. Furthermore, the study of orbits and revolutions of satellites is driven by the study of circles. The first part of this chapter, 'Conic Sections', deals with various aspects of circles. These circles are formed when a plane intersects a cone in a particular way. Thereafter advanced concepts of conic sections are discussed.
In this NCERT exercise, 10.1 of Chapter 10, we have dealt with the questions related to circles, along with the representation of the equation of a circle in different forms. The exercise also involves solving questions such as determining the equation of a circle when its centre and radius are provided or identifying the centre and radius from a given equation of a circle. To get conceptual clarity on this topic, students are advised to solve the given problems on their own in the first go without seeing the solutions, thereafter referring to the solutions provided. Students can also access complete NCERT Solutions from Class 6 to Class 12 provided here in an elaborate manner.
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centre (0,2) and radius 2
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
So Given Here
$(h,k)=(0,2)$
AND $r=2$
So the equation of the circle is:
, $(x-0)^2+(y-2)^2=2^2$
$x^2+y^2-4y+4=4$
$x^2+y^2-4y=0$
Question:2 Find the equation of the circle with
centre (–2,3) and radius 4
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
So Given Here
$(h,k)=(-2,3)$
AND $r=4$
So the equation of the circle is:
, $(x-(-2))^2+(y-3)^2=4^2$
$x^2+4x+4+y^2-6y+9=16$
$x^2+y^2+4x-6y-3=0$
Question:3 Find the equation of the circle with
centre $\left(\frac{1}{2},\frac{1}{4} \right )$ and radius $\frac{1}{12}$
Answer:
As we know,
The equation of the circle with center ( h, k) and radius r is give by ;
$(x-h)^2+(y-k)^2=r^2$
So Given Here
$(h,k)=\left ( \frac{1}{2},\frac{1}{4} \right )$
AND
$r=\frac{1}{12}$
So the equation of circle is:
, $\left ( x-\frac{1}{2}\right )^2+\left ( y-\frac{1}{4}\right )^2=\left ( \frac{1}{12}\right )^2$
$x^2-x+\frac{1}{4}+y^2-\frac{1}{2}y+\frac{1}{16}=\frac{1}{144}$
$x^2+y^2-x-\frac{1}{2}y-\frac{11}{36}=0$
$36x^2+36y^2-36x-18y-11=0$
Question:4 Find the equation of the circle with
centre (1,1) and radius $\sqrt2$
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
So Given Here
$(h,k)=(1,1)$
AND $r=\sqrt{2}$
So the equation of the circle is:
, $(x-1)^2+(y-1)^2=(\sqrt{2})^2$
$x^2-2x+1+y^2-2y+1=2$
$x^2+y^2-2x-2y=0$
Question:5 Find the equation of the circle with
centre $(-a,-b)$ and radius $\sqrt{a^2 - b^2}$
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
So Given Here
$(h,k)=(-a,-b)$
AND $r=\sqrt{a^2-b^2}$
So the equation of the circle is:
, $(x-(-a))^2+(y-(-b))^2=(\sqrt{a^2-b^2})^2$
$x^2+2ax+a^2+y^2+2by+b^2=a^2-b^2$
$x^2+y^2+2ax+2by+2b^2=0$
Question:6 Find the centre and radius of the circles.
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given here
$(x+5)^2 + (y-3)^2 = 36$
Can also be written in the form
$(x-(-5))^2 + (y-3)^2 = 6^2$
So, from comparing, we can see that
$r=6$
Hence the Radius of the circle is 6.
Question:7 Find the centre and radius of the circles.
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given here
$x^2 + y^2 -4x - 8y - 45 = 0$
Can also be written in the form
$(x-2)^2 + (y-4)^2 =(\sqrt{65})^2$
So, from comparing, we can see that
$r=\sqrt{65}$
Hence the Radius of the circle is $\sqrt{65}$.
Question:8 Find the centre and radius of the circles.
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given here
$x^2 + y^2 -8x +10y -12 = 0$
Can also be written in the form
$(x-4)^2 + (y-(-5))^2 = (\sqrt{53})^2$
So, from comparing, we can see that
$r=\sqrt{53}$
Hence the radius of the circle is $\sqrt{53}$.
Question:9 Find the centre and radius of the circles.
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given here
$2x^2 + 2y^2 - x = 0$
Can also be written in the form
$\left ( x-\frac{1}{4}\right )^2 + \left ( y-0 \right )^2 = \left ( \frac{1}{4} \right )^2$
So, from comparing, we can see that
$r=\frac{1}{4}$
Hence Center of the circle is the $\left ( \frac{1}{4},0\right )$Radius of the circle is $\frac{1}{4}$.
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given Here,
Condition 1: the circle passes through points (4,1) and (6,5)
$(4-h)^2+(1-k)^2=r^2$
$(6-h)^2+(5-k)^2=r^2$
Here,
$(4-h)^2+(1-k)^2=(6-h)^2+(5-k)^2$
$(4-h)^2-(6-h)^2+(1-k)^2-(5-k)^2=0$
$(-2)(10-2h)+(-4)(6-2k)=0$
$-20+4h-24+8k=0$
$4h+8k=44$
Now, Condition 2:centre is on the line $4x + y = 16$.
$4h+k=16$
From condition 1 and condition 2
$h=3,\:k=4$
Now lets substitute this value of h and k in condition 1 to find out r
$(4-3)^2+(1-4)^2=r^2$
$1+9=r^2$
$r=\sqrt{10}$
So now, the Final Equation of the circle is
$(x-3)^2+(y-4)^2=(\sqrt{10})^2$
$x^2-6x+9+y^2-8y+16=10$
$x^2+y^2-6x-8y+15=0$
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
Given Here,
Condition 1: the circle passes through points (2,3) and (–1,1)
$(2-h)^2+(3-k)^2=r^2$
$(-1-h)^2+(1-k)^2=r^2$
Here,
$(2-h)^2+(3-k)^2=(-1-h)^2+(1-k)^2$
$(2-h)^2-(-1-h)^2+(3-k)^2-(1-k)^2=0$
$(3)(1-2h)+(2)(4-2k)=0$
$3-6h+8-4k=0$
$6h+4k=11$
Now, Condition 2: centre is on the line.$x - 3y - 11 = 0$
$h-3k=11$
From condition 1 and condition 2
$h=\frac{7}{2},\:k=\frac{-5}{2}$
Now let's substitute this value of h and k in condition 1 to find out $r$
$\left ( 2-\frac{7}{2}\right )^2+\left (3+\frac{5}{2}\right )^2=r^2$
$\frac{9}{4}+\frac{121}{4}=r^2$
$r^2=\frac{130}{4}$
So now, the Final Equation of the circle is
$\left(x-\frac{7}{2}\right )^2+\left(y+\frac{5}{2}\right)^2=\frac{130}{4}$
$x^2-7x+\frac{49}{4}+y^2+5y+\frac{25}{4}=\frac{130}{4}$
$x^2+y^2-7x+5y-\frac{56}{4}=0$
$x^2+y^2-7x+5y-14=0$
Answer:
As we know,
The equation of the circle with centre ( h, k) and radius r is given by ;
$(x-h)^2+(y-k)^2=r^2$
So let the circle be,
$(x-h)^2+(y-k)^2=r^2$
Since it's radius is 5 and its centre lies on x-axis,
$(x-h)^2+(y-0)^2=5^2$
And Since it passes through the point (2,3).
$(2-h)^2+(3-0)^2=5^2$
$(2-h)^2=25-9$
$(2-h)^2=16$
$(2-h)=4\:or\:(2-h)=-4$
$h=-2\: or\;6$
When $h=-2\:$ ,The equation of the circle is :
$(x-(-2))^2+(y-0)^2=5^2$
$x^2+4x+4+y^2=25$
$x^2+y^2+4x-21=0$
When $h=6$ The equation of the circle is :
$(x-6)^2+(y-0)^2=5^2$
$x^2-12x+36+y^2=25$
$x^2+y^2-12x+11=0$
Question:13 Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes.
Answer:
Let the equation of circle be,
$(x-h)^2+(y-k)^2=r^2$
Now since this circle passes through (0,0)
$(0-h)^2+(0-k)^2=r^2$
$h^2+k^2=r^2$
Now, this circle makes an intercept of a and b on the coordinate axes.it means circle passes through the point (a,0) and (0,b)
So,
$(a-h)^2+(0-k)^2=r^2$
$a^2-2ah+h^2+k^2=r^2$
$a^2-2ah=0$
$a(a-2h)=0$
$a=0\:or\:a-2h=0$
Since $a\neq0\:so\:a-2h=0$
$h=\frac{a}{2}$
Similarly,
$(0-h)^2+(b-k)^2=r^2$
$h^2+b^2-2bk+k^2=r^2$
$b^2-2bk=0$
$b(b-2k)=0$
Since b is not equal to zero.
$k=\frac{b}{2}$
So Final equation of the Circle ;
$\left ( x-\frac{a}{2} \right )^2+\left ( y-\frac{b}{}2 \right )^2=\left ( \frac{a}{2} \right )^2+\left ( \frac{b}{2} \right )^2$
$x^2-ax+\frac{a^2}{4}+y^2-bx+\frac{b^2}{4}=\frac{a^2}{4}+\frac{b^2}{4}$
$x^2+y^2-ax-bx=0$
Question:14 Find the equation of a circle with centre (2,2) and passes through the point (4,5).
Answer:
Let the equation of circle be :
$(x-h)^2+(y-k)^2=r^2$
Now, since the centre of the circle is (2,2), our equation becomes
$(x-2)^2+(y-2)^2=r^2$
Now, Since this passes through the point (4,5)
$(4-2)^2+(5-2)^2=r^2$
$4+9=r^2$
$r^2=13$
Hence The Final equation of the circle becomes
$(x-2)^2+(y-2)^2=13$
$x^2-4x+4+y^2-4y+4=13$
$x^2+y^2-4x-4y-5=0$
Question:15 Does the point (–2.5, 3.5) lie inside, outside or on the circle $x^2 + y^2 = 25$?
Answer:
Given, a circle
$x^2 + y^2 = 25$
As we can see center of the circle is ( 0,0)
Now the distance between (0,0) and (–2.5, 3.5) is
$d=\sqrt{(-2.5-0)^2+(3.5-0)^2}$
$d=\sqrt{6.25+12.25}$
$d=\sqrt{18.5}\approx 4.3$$d=\sqrt{18.5}\approx 4.3<5$
Since distance between the given point and center of the circle is less than radius of the circle, the point lie inside the circle.
Also read,
Conic Section Exercise 10.2 |
Conic Section Exercise 10.3 |
Conic Section Exercise 10.4 |
Conic Section Miscellaneous Exercise |
1. Introduction to Conic Sections: In this section, a brief idea about conic sections is provided. Various shapes related to conic sections such as circle, ellipse, parabola, hyperbola is dicussed having more emphasis upon circle..
2. Definition of a Circle: A circle is defined as the set of all points in a plane that are equidistant from a fixed point (called the centre). The fixed distance is called the radius.
3. Application-Based Questions: This involves the questions based on:
Identifying circle parameters.
Verifying if a point lies on a circle.
Finding missing values (like radius or coordinates of the centre).
Real-life applications involving the equation of a circle.
Follow the links to get your hands on subject-wise NCERT textbooks and exemplar solutions to ace your exam preparation.
NCERT Exemplar Solutions for Class 11 Maths |
NCERT Exemplar Solutions for Class 11 Physics |
NCERT Exemplar Solutions for Class 11 Chemistry |
NCERT Exemplar Solutions for Class 11 Biology |
Also see-
Happy learning!!!
Frequently Asked Questions (FAQs)
The curves obtained as intersections of a plane with a double-napped right circular cone are called conics.
The curve obtained from the set of all points in a plane that are equidistant from a fixed point in the plane is called a circle.
The distance between the centre to a point on the circle is called the radius of the circle.
Equation of the circle centre at (0,0) and radius 3 => x^2 + y^2 = 3^2
x^2 + y^2 = 9
Equation of the circle centre at (1,2) and radius 2 => (x-1)^2 + (y-2)^2 = 2^2
x^2 -2x + 1 + y^2 - 4y + 4= 4
x^2 -2x + y^2 - 4y + 1= 0
Equation of the circle => x^2 + y^2 + 4x + 6y – 12 = 0
(x+2)^2 + (y+3)^2 – 25 = 0
(x+2)^2 + (y+3)^2 = 5^2
Centre(-2,-3) and radius = 5.
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