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The equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes is:
- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The equation of the circle passing through (0, 0) and making intercept...
The circle intercept the co-ordinate axes at a and b. it means x - intercept at ( a, 0) and y-intercept at (0, b) .
Now, we observed that circle passes through points (0, 0) , (a, 0) and (0, b) .
we also know, General equation of circle is
x² + y² + 2gx + 2fy + C = 0
when point (0,0)
(0)² + (0)² + 2g(0) + 2f(0) + C = 0
0 + 0 + 0 + 0 + C = 0
C = 0 -------(1)
when point (a,0)
(a)² + (0)² + 2g(a) + 2f(0) + C = 0
a² + 2ag + C = 0
from equation (1)
a² + 2ag = 0
a(a + 2g) = 0
g = -a/2
when point ( 0, b)
(0)² + (b)² + 2g(0) + 2f(b) + C = 0
b² + 2fb + C = 0
f = -b/2
Now, equation of circle is
x² + y² + 2x(-a/2) + 2y(-b/2) + 0 = 0 { after putting values of g, f and C }
x² + y² - ax - by = 0
As we know that, a=2, b=4
x^2 + y^2 - 2x - 4y = 0
Now, we observed that circle passes through points (0, 0) , (a, 0) and (0, b) .
we also know, General equation of circle is
x² + y² + 2gx + 2fy + C = 0
when point (0,0)
(0)² + (0)² + 2g(0) + 2f(0) + C = 0
0 + 0 + 0 + 0 + C = 0
C = 0 -------(1)
when point (a,0)
(a)² + (0)² + 2g(a) + 2f(0) + C = 0
a² + 2ag + C = 0
from equation (1)
a² + 2ag = 0
a(a + 2g) = 0
g = -a/2
when point ( 0, b)
(0)² + (b)² + 2g(0) + 2f(b) + C = 0
b² + 2fb + C = 0
f = -b/2
Now, equation of circle is
x² + y² + 2x(-a/2) + 2y(-b/2) + 0 = 0 { after putting values of g, f and C }
x² + y² - ax - by = 0
As we know that, a=2, b=4
x^2 + y^2 - 2x - 4y = 0
Most Upvoted Answer
The equation of the circle passing through (0, 0) and making intercept...
To find the equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes, we can use the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Step 1: Finding the center of the circle
Since the circle passes through (0, 0), we can substitute these coordinates into the equation:
(0 - h)^2 + (0 - k)^2 = r^2
Simplifying this equation gives us:
h^2 + k^2 = r^2 ----(1)
Step 2: Finding the radius of the circle
The circle makes intercepts 2 and 4 on the coordinate axes. This means that the distance between the center of the circle and the x-axis is equal to 2, and the distance between the center of the circle and the y-axis is equal to 4.
Using the distance formula, we can write the following equations:
√(h^2 + (k - 4)^2) = 2 ----(2)
√((h - 2)^2 + k^2) = 4 ----(3)
Squaring equation (2), we get:
h^2 + (k - 4)^2 = 4 ----(4)
Squaring equation (3), we get:
(h - 2)^2 + k^2 = 16 ----(5)
Expanding equation (4) gives us:
h^2 + k^2 - 8k + 16 = 4 ----(6)
Expanding equation (5) gives us:
h^2 - 4h + 4 + k^2 = 16 ----(7)
Combining equations (6) and (7), we get:
-4h - 8k + 4 = 0 ----(8)
Step 3: Substituting the values into the equation of the circle
Substituting the values of h^2 + k^2 from equation (1) into equation (8), we get:
r^2 - 4h - 8k + 4 = 0
Rearranging this equation, we get:
r^2 = 4h + 8k - 4 ----(9)
Substituting the values of h^2 + k^2 from equation (1) into equation (9), we get:
r^2 = 4r^2 - 4
Simplifying this equation gives us:
3r^2 = 4
Therefore, r^2 = 4/3
Step 4: Writing the final equation of the circle
Substituting the values of h^2 + k^2 and r^2 into the general equation of a circle, we get:
(x - 0)^2 + (y - 0)^2 = 4/3
Simplifying this equation gives us:
x^2 + y^2 = 4/3
So, the correct equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes is:
x^2 + y^2 - 4/3 =
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Step 1: Finding the center of the circle
Since the circle passes through (0, 0), we can substitute these coordinates into the equation:
(0 - h)^2 + (0 - k)^2 = r^2
Simplifying this equation gives us:
h^2 + k^2 = r^2 ----(1)
Step 2: Finding the radius of the circle
The circle makes intercepts 2 and 4 on the coordinate axes. This means that the distance between the center of the circle and the x-axis is equal to 2, and the distance between the center of the circle and the y-axis is equal to 4.
Using the distance formula, we can write the following equations:
√(h^2 + (k - 4)^2) = 2 ----(2)
√((h - 2)^2 + k^2) = 4 ----(3)
Squaring equation (2), we get:
h^2 + (k - 4)^2 = 4 ----(4)
Squaring equation (3), we get:
(h - 2)^2 + k^2 = 16 ----(5)
Expanding equation (4) gives us:
h^2 + k^2 - 8k + 16 = 4 ----(6)
Expanding equation (5) gives us:
h^2 - 4h + 4 + k^2 = 16 ----(7)
Combining equations (6) and (7), we get:
-4h - 8k + 4 = 0 ----(8)
Step 3: Substituting the values into the equation of the circle
Substituting the values of h^2 + k^2 from equation (1) into equation (8), we get:
r^2 - 4h - 8k + 4 = 0
Rearranging this equation, we get:
r^2 = 4h + 8k - 4 ----(9)
Substituting the values of h^2 + k^2 from equation (1) into equation (9), we get:
r^2 = 4r^2 - 4
Simplifying this equation gives us:
3r^2 = 4
Therefore, r^2 = 4/3
Step 4: Writing the final equation of the circle
Substituting the values of h^2 + k^2 and r^2 into the general equation of a circle, we get:
(x - 0)^2 + (y - 0)^2 = 4/3
Simplifying this equation gives us:
x^2 + y^2 = 4/3
So, the correct equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes is:
x^2 + y^2 - 4/3 =
Community Answer
The equation of the circle passing through (0, 0) and making intercept...
Simple put y is equal to 0 for x inttercept and x is equal to 0 for y intercept in each equation.
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The equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes is:a)b)c)d)Correct answer is option 'C'. Can you explain this answer?
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