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Consider the following statements in respect of circles x2 + y2 – 2x – 2y = 0 and x2 + y2 = 1
I. The radius of the first circle is twice that of the second circle.
II. Both the circles pass through the origin.
II. Both the circles pass through the origin.
- a)Only I
- b)Only II
- c)Both I and II
- d)Neither I nor II
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Consider the following statements in respect of circles x2 + y2 2x 2...
Radius of the circles
To determine the radius of the circles, we can rewrite the equations of the circles in standard form.
The first circle equation, x^2 + y^2 + 2x + 2y = 0, can be rearranged as (x+1)^2 + (y+1)^2 = 2. Comparing this equation with the standard form equation (x-a)^2 + (y-b)^2 = r^2, we can see that the center of the circle is (-1, -1) and the radius is sqrt(2).
The second circle equation, x^2 + y^2 = 1, has the standard form equation with the center at the origin (0, 0) and a radius of 1.
Comparing the radii
Now, let's compare the radii of the two circles.
The radius of the first circle is sqrt(2), while the radius of the second circle is 1.
Since sqrt(2) is not twice the value of 1, we can conclude that statement I is false.
Passing through the origin
To determine if both circles pass through the origin, we can substitute (0, 0) into the equations of the circles.
For the first circle, substituting x=0 and y=0 gives us 0^2 + 0^2 + 2(0) + 2(0) = 0. Thus, the origin is a solution to the first circle equation.
For the second circle, substituting x=0 and y=0 gives us 0^2 + 0^2 = 1. Since this equation is not satisfied, the origin is not a solution to the second circle equation.
Therefore, only the first circle passes through the origin, and statement II is false.
Conclusion
Based on our analysis, we can conclude that neither statement I nor statement II is correct. The correct answer is option D, "Neither I nor II."
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Community Answer
Consider the following statements in respect of circles x2 + y2 2x 2...
Statement I: The radius of the first circle is twice that of the second circle.
To determine the radius of each circle, we need to rewrite the equations in standard form:
First circle: x^2 + y^2 - 2x - 2y = 0
Rearranging the terms, we get:
(x^2 - 2x) + (y^2 - 2y) = 0
Completing the square, we add and subtract the square of half the coefficient of x and y respectively:
(x^2 - 2x + 1) + (y^2 - 2y + 1) = 1 + 1
(x - 1)^2 + (y - 1)^2 = 2
Comparing this with the standard form of a circle, (x - h)^2 + (y - k)^2 = r^2, we can see that the radius of the first circle is √2.
Second circle: x^2 + y^2 = 1
Comparing this with the standard form, we can see that the radius of the second circle is 1.
Therefore, the radius of the first circle is not twice that of the second circle. Hence, statement I is false.
Statement II: Both the circles pass through the origin.
To determine if a circle passes through the origin, we substitute x = 0 and y = 0 into the equation of the circle.
For the first circle:
(0 - 1)^2 + (0 - 1)^2 = 2
1 + 1 = 2
For the second circle:
0^2 + 0^2 = 1
Both equations satisfy the condition, so both the circles pass through the origin. Hence, statement II is true.
Therefore, only statement II is correct. The correct answer is option B: Only II.
To determine the radius of each circle, we need to rewrite the equations in standard form:
First circle: x^2 + y^2 - 2x - 2y = 0
Rearranging the terms, we get:
(x^2 - 2x) + (y^2 - 2y) = 0
Completing the square, we add and subtract the square of half the coefficient of x and y respectively:
(x^2 - 2x + 1) + (y^2 - 2y + 1) = 1 + 1
(x - 1)^2 + (y - 1)^2 = 2
Comparing this with the standard form of a circle, (x - h)^2 + (y - k)^2 = r^2, we can see that the radius of the first circle is √2.
Second circle: x^2 + y^2 = 1
Comparing this with the standard form, we can see that the radius of the second circle is 1.
Therefore, the radius of the first circle is not twice that of the second circle. Hence, statement I is false.
Statement II: Both the circles pass through the origin.
To determine if a circle passes through the origin, we substitute x = 0 and y = 0 into the equation of the circle.
For the first circle:
(0 - 1)^2 + (0 - 1)^2 = 2
1 + 1 = 2
For the second circle:
0^2 + 0^2 = 1
Both equations satisfy the condition, so both the circles pass through the origin. Hence, statement II is true.
Therefore, only statement II is correct. The correct answer is option B: Only II.
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