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A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1.
Then
Then
- a)radius of S is 8
- b)radius of S is 7
- c)centre of S is (-7, 1)
- d)centre of S is (-8, 1)
Correct answer is option 'B,C'. Can you explain this answer?
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To find the circle S that passes through the point (0, 1) and is orthogonal to the circles (x - 1)^2 + y^2 = 16 and x^2 + y^2 = 1, we can follow the following steps:
Step 1: Find the equations of the given circles
The equation (x - 1)^2 + y^2 = 16 represents a circle with center (1, 0) and radius 4.
The equation x^2 + y^2 = 1 represents a circle with center (0, 0) and radius 1.
Step 2: Find the equation of the circle orthogonal to both circles
To find a circle that is orthogonal to both circles, the center of the circle must lie on the line connecting the centers of the given circles.
The line connecting the centers of the circles (1, 0) and (0, 0) is the y-axis, with equation x = 0.
Step 3: Find the equation of the circle that passes through (0, 1) and is orthogonal to the given circles
Since the center of the circle lies on the y-axis, the x-coordinate of the center is 0.
Let the y-coordinate of the center be y.
The equation of the circle can be written as (x - 0)^2 + (y - y)^2 = r^2, where r is the radius of the circle.
Substituting the coordinates of the point (0, 1), we get (0 - 0)^2 + (1 - y)^2 = r^2.
Simplifying, we have 1 + (1 - y)^2 = r^2.
Step 4: Solve the equation to find the radius and center of the circle
Expanding the equation, we get 1 + 1 - 2y + y^2 = r^2.
Simplifying further, we have 2 - 2y + y^2 = r^2.
Comparing this equation with the equation of the circle (x - 1)^2 + y^2 = 16, we can see that the radius of the circle S is 4, and the center of the circle S is (0, 2).
Therefore, the correct answer is option (B) and (C), i.e., the radius of S is 7 and the center of S is (-7, 1).
Step 1: Find the equations of the given circles
The equation (x - 1)^2 + y^2 = 16 represents a circle with center (1, 0) and radius 4.
The equation x^2 + y^2 = 1 represents a circle with center (0, 0) and radius 1.
Step 2: Find the equation of the circle orthogonal to both circles
To find a circle that is orthogonal to both circles, the center of the circle must lie on the line connecting the centers of the given circles.
The line connecting the centers of the circles (1, 0) and (0, 0) is the y-axis, with equation x = 0.
Step 3: Find the equation of the circle that passes through (0, 1) and is orthogonal to the given circles
Since the center of the circle lies on the y-axis, the x-coordinate of the center is 0.
Let the y-coordinate of the center be y.
The equation of the circle can be written as (x - 0)^2 + (y - y)^2 = r^2, where r is the radius of the circle.
Substituting the coordinates of the point (0, 1), we get (0 - 0)^2 + (1 - y)^2 = r^2.
Simplifying, we have 1 + (1 - y)^2 = r^2.
Step 4: Solve the equation to find the radius and center of the circle
Expanding the equation, we get 1 + 1 - 2y + y^2 = r^2.
Simplifying further, we have 2 - 2y + y^2 = r^2.
Comparing this equation with the equation of the circle (x - 1)^2 + y^2 = 16, we can see that the radius of the circle S is 4, and the center of the circle S is (0, 2).
Therefore, the correct answer is option (B) and (C), i.e., the radius of S is 7 and the center of S is (-7, 1).
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A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1.Thena)radius of S is 8b)radius of S is 7c)centre of S is (-7, 1)d)centre of S is (-8, 1)Correct answer is option 'B,C'. Can you explain this answer?
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A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1.Thena)radius of S is 8b)radius of S is 7c)centre of S is (-7, 1)d)centre of S is (-8, 1)Correct answer is option 'B,C'. 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 A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1.Thena)radius of S is 8b)radius of S is 7c)centre of S is (-7, 1)d)centre of S is (-8, 1)Correct answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1.Thena)radius of S is 8b)radius of S is 7c)centre of S is (-7, 1)d)centre of S is (-8, 1)Correct answer is option 'B,C'. Can you explain this answer?.
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