
If the circle cuts the three circles and at the extremities of their diameter, then which of the following is/are true?
(a)
(b)
(c)
(d)
Answer
499.8k+ views
Hint: In order to solve this problem we need to use the concept of the orthogonal circle, orthogonal circles are those circles which intersect at right angles. We need to compare the coefficients of given circles with the general equation of the circle then use the condition of orthogonal of circles. The formula for condition of orthogonality:
Complete step-by-step answer:
Orthogonal circles:
When the first circle intersects the three circles at the extremities of the diameter, then they are orthogonal to each other.
Given that the equations of three circles are
and
The center of the first circle , is at .
For center of second circle, , we have:
and
Centre of the second circle, , is at .
Similarly for the center of third circle, , we have:
and
Centre of the second circle, , is at .
The circle which cuts these circles is given by:
Then its center is:
Constant term of circle is
Constant term of circle is
Constant term of circle is
Constant term of the circle which cuts these circle i.e. is
By using the concept of the orthogonal circle, we’ll get:
For the circle
Thus, option (a) is incorrect.
Similarly using the concept of the orthogonal for the circle , we’ll get
Rewrite the equation after simplification
Substitute the value of c in equation , we’ll get:
Use the concept of the orthogonal circle for the circle , we’ll get:
Rewrite the equation after simplification:
Simplifying the equation, we have:
Substitute the value of the f in the equation :
Simplifying the equation we’ll get:
Substitute the value of g back in equation :
Simplifying it further, we’ll get:
Therefore, the product of f and g is:
Therefore option (b) is incorrect.
Comparing the values of and , we’ll get:
and
So we get:
Hence, option (d) is the correct.
Now, comparing the value of and , we have:
Similarly we have:
So we get:
Thus, option (c) is also incorrect.
Hence, only option (d) is correct.
Note: The general equation of circle is given as:
From this general equation, we get the center of the circle as and its radius is .
For a two degree equation in and to be circle, the coefficients of and must be the same.
Complete step-by-step answer:
Orthogonal circles:
When the first circle intersects the three circles at the extremities of the diameter, then they are orthogonal to each other.
Given that the equations of three circles are
The center of the first circle
For center of second circle,
and
Centre of the second circle,
Similarly for the center of third circle,
and
Centre of the second circle,
The circle which cuts these circles is given by:
Then its center is:
Constant term of circle
Constant term of circle
Constant term of circle
Constant term of the circle which cuts these circle i.e.
By using the concept of the orthogonal circle, we’ll get:
For the circle
Thus, option (a) is incorrect.
Similarly using the concept of the orthogonal for the circle
Rewrite the equation after simplification
Substitute the value of c in equation
Use the concept of the orthogonal circle for the circle
Rewrite the equation after simplification:
Simplifying the equation, we have:
Substitute the value of the f in the equation
Simplifying the equation we’ll get:
Substitute the value of g back in equation
Simplifying it further, we’ll get:
Therefore, the product of f and g is:
Therefore option (b) is incorrect.
Comparing the values of
and
So we get:
Hence, option (d) is the correct.
Now, comparing the value of
Similarly we have:
So we get:
Thus, option (c) is also incorrect.
Hence, only option (d) is correct.
Note: The general equation of circle is given as:
From this general equation, we get the center of the circle as
For a two degree equation in
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