
Find the equation of the circle which touches both axes in the first quadrant and whose radius is a
A )$\quad x^{2}+y^{2}-2 a x-2 a y+a^{2}=0$
B )$\quad x^{2}+y^{2}-2 a x+2 a y-a^{2}=0$
C )$\quad x^{2}+y^{2}+2 a x-2 a y+a^{2}=0$
D ) None of the above
Answer
161.4k+ views
Hint:We have the general equation of circle. And it is given that the circle touches both axes in the first quadrant. So we can find the general equation of the given circle by substituting the a value. The radius of a circle is one of its most important features It is the separation between a point on a circle's edge and its center.
Complete step by step Solution:
Given,
In the first quadrant, a circle with radius a contacts both axes, hence its center will be $(a, a)$.

Therefore the required equation is
$\Rightarrow(x-a)^{2}+(y-a)^{2}=a^{2}$
$\Rightarrow x^{2}+a^{2}-2 a x+y^{2}+a^{2}-2 a y=a^{2}$
$\Rightarrow x^{2}+y^{2}-2 a x-2 a y+2 a^{2}-a^{2}=0$
$\Rightarrow {{x}^{2}}+{{y}^{2}}-2ax-2ay+{{a}^{2}}=0$
Therefore, the correct option is A.
Additional information:
The radius of a circle is the length of the straight line that connects the center to any point on its circumference. Because a circle's circumference can contain an endless number of points, a circle can have more than one radius. This indicates that a circle has an endless number of radii and that each radius is equally spaced from the circle's center. When the radius's length varies, the circle's size also changes.
Note: The equation for a circle has the generic form: ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$. The coordinates of the circle's center and radius are found using this general form, where g, f, and c are constants. The general form of the equation of a circle makes it difficult to identify any significant properties about any specific circle, in contrast to the standard form, which is simpler to comprehend. So, to quickly change from the generic form to the standard form, we will use the completing square formula.
Complete step by step Solution:
Given,
In the first quadrant, a circle with radius a contacts both axes, hence its center will be $(a, a)$.

Therefore the required equation is
$\Rightarrow(x-a)^{2}+(y-a)^{2}=a^{2}$
$\Rightarrow x^{2}+a^{2}-2 a x+y^{2}+a^{2}-2 a y=a^{2}$
$\Rightarrow x^{2}+y^{2}-2 a x-2 a y+2 a^{2}-a^{2}=0$
$\Rightarrow {{x}^{2}}+{{y}^{2}}-2ax-2ay+{{a}^{2}}=0$
Therefore, the correct option is A.
Additional information:
The radius of a circle is the length of the straight line that connects the center to any point on its circumference. Because a circle's circumference can contain an endless number of points, a circle can have more than one radius. This indicates that a circle has an endless number of radii and that each radius is equally spaced from the circle's center. When the radius's length varies, the circle's size also changes.
Note: The equation for a circle has the generic form: ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$. The coordinates of the circle's center and radius are found using this general form, where g, f, and c are constants. The general form of the equation of a circle makes it difficult to identify any significant properties about any specific circle, in contrast to the standard form, which is simpler to comprehend. So, to quickly change from the generic form to the standard form, we will use the completing square formula.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
