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One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal to
- a)13/2
- b)15/2
- c)17/2
- d)none
Correct answer is option 'A'. Can you explain this answer?
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One circle has a radius of 5 and its center at (0, 5). A second circle...
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One circle has a radius of 5 and its center at (0, 5). A second circle...
Given:
- Circle 1 has a radius of 5 and its center at (0, 5).
- Circle 2 has a radius of 12 and its center at (12, 0).
To find:
- The length of the radius of a third circle that passes through the center of the second circle and both points of intersection of the first and second circles.
Explanation:
Step 1: Find the equations of the two circles:
- The equation of Circle 1 with center (0, 5) and radius 5 is: (x - 0)^2 + (y - 5)^2 = 5^2
- The equation of Circle 2 with center (12, 0) and radius 12 is: (x - 12)^2 + (y - 0)^2 = 12^2
Step 2: Find the points of intersection:
- To find the points of intersection, we need to solve the system of equations formed by the two circle equations.
- Substituting the equation of Circle 1 into Circle 2, we get: (x - 12)^2 + (y - 0)^2 = 12^2
- Expanding and simplifying the equation, we get: x^2 - 24x + 144 + y^2 = 144
- Simplifying further, we get: x^2 + y^2 - 24x = 0
- Rearranging the equation, we get: x^2 - 24x + y^2 = 0
Step 3: Find the coordinates of the intersection points:
- We can solve the quadratic equation x^2 - 24x + y^2 = 0 to find the x-coordinates of the intersection points.
- The discriminant of the quadratic equation is: D = (-24)^2 - 4(1)(y^2)
- Since the two circles intersect, the discriminant must be greater than zero.
- Solving for D > 0, we get: 576 - 4y^2 > 0
- Simplifying further, we get: y^2 < />
- Taking the square root on both sides, we get: -12 < y="" />< />
Step 4: Find the length of the radius of the third circle:
- The third circle passes through the center of Circle 2, which is (12, 0).
- Let the coordinates of one of the intersection points be (x1, y1) and the coordinates of the other intersection point be (x2, y2).
- Using the distance formula, the length of the radius of the third circle is: sqrt((x1 - 12)^2 + (y1 - 0)^2)
Step 5: Substitute the values and find the answer:
- Since the third circle passes through the center of Circle 2, (x1, y1) = (12, 0).
- Substituting the values into the distance formula, we get: sqrt((12 - 12)^2 + (0 - 0)^2) = sqrt(0 + 0) = 0
Conclusion:
- The length of the
- Circle 1 has a radius of 5 and its center at (0, 5).
- Circle 2 has a radius of 12 and its center at (12, 0).
To find:
- The length of the radius of a third circle that passes through the center of the second circle and both points of intersection of the first and second circles.
Explanation:
Step 1: Find the equations of the two circles:
- The equation of Circle 1 with center (0, 5) and radius 5 is: (x - 0)^2 + (y - 5)^2 = 5^2
- The equation of Circle 2 with center (12, 0) and radius 12 is: (x - 12)^2 + (y - 0)^2 = 12^2
Step 2: Find the points of intersection:
- To find the points of intersection, we need to solve the system of equations formed by the two circle equations.
- Substituting the equation of Circle 1 into Circle 2, we get: (x - 12)^2 + (y - 0)^2 = 12^2
- Expanding and simplifying the equation, we get: x^2 - 24x + 144 + y^2 = 144
- Simplifying further, we get: x^2 + y^2 - 24x = 0
- Rearranging the equation, we get: x^2 - 24x + y^2 = 0
Step 3: Find the coordinates of the intersection points:
- We can solve the quadratic equation x^2 - 24x + y^2 = 0 to find the x-coordinates of the intersection points.
- The discriminant of the quadratic equation is: D = (-24)^2 - 4(1)(y^2)
- Since the two circles intersect, the discriminant must be greater than zero.
- Solving for D > 0, we get: 576 - 4y^2 > 0
- Simplifying further, we get: y^2 < />
- Taking the square root on both sides, we get: -12 < y="" />< />
Step 4: Find the length of the radius of the third circle:
- The third circle passes through the center of Circle 2, which is (12, 0).
- Let the coordinates of one of the intersection points be (x1, y1) and the coordinates of the other intersection point be (x2, y2).
- Using the distance formula, the length of the radius of the third circle is: sqrt((x1 - 12)^2 + (y1 - 0)^2)
Step 5: Substitute the values and find the answer:
- Since the third circle passes through the center of Circle 2, (x1, y1) = (12, 0).
- Substituting the values into the distance formula, we get: sqrt((12 - 12)^2 + (0 - 0)^2) = sqrt(0 + 0) = 0
Conclusion:
- The length of the
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One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer?
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One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer?.
One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer?.
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One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at (12, 0). The length of a radius of a third circle which passes through the center of the second circle and both points of intersection of the first 2 circles, is equal toa)13/2b)15/2c)17/2d)noneCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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