
If represents a circle, then h is:
(a) (b) (c) (d)
Answer
530.7k+ views
Hint: Think back to the general form of the equation of a circle. It is . Now see if there’s any term that contains an xy. Proceed from here.
The general equation of any conic section can be written as :
.
Comparing equation , we find :
.
For this conic to be the equation of a circle, there are certain conditions that this general equation needs to follow. They are :
1. The value of . (This makes sure that the conic is an ellipse).
2. Since a circle is just a special case of an ellipse, having both the lengths of the major and minor axes as equal, another condition for the conic to be a circle is : and .
Now, we will check for the conditions to be satisfied one by one.
Condition (1) says that .
Substituting for in the equation , we get :
Therefore, can be any value between and according to the first condition.
However, satisfying the first condition just makes the conic represent an ellipse, rather than a circle.
The next condition to be satisfied is condition (2), which states that for the conic to be a circle, and .
Substituting for , we get :
which is always true.
And,
Thus, for the second condition to be satisfied, the only possible value of can be .
also satisfies the interval we got from the first condition. It does lie between and .
Hence, should be equal to for the equation to represent a circle.
Correct answer is option (b)
Note: The general equation listed in this sum is an equation that is true for all conic sections. Imposing restrictions on this general form gives us different types of conics, like circles, ellipses, parabolas, etc. Just like we solved for the equation to be a circle right now, it can also be made to represent a parabola or any other conic simply by tweaking the variables.
Another way of approaching this problem would be to simply compare the equation given to the general form of a circle which is : .
As we can see here, there’s no term containing both and in the general equation of a circle. Hence, on comparing we can directly reach the conclusion that the coefficient of in the equation given. Even then, we would have arrived at the same result.
The general equation of any conic section can be written as :
Comparing equation
For this conic to be the equation of a circle, there are certain conditions that this general equation needs to follow. They are :
1. The value of
2. Since a circle is just a special case of an ellipse, having both the lengths of the major and minor axes as equal, another condition for the conic to be a circle is :
Now, we will check for the conditions to be satisfied one by one.
Condition (1) says that
Substituting for
Therefore,
However, satisfying the first condition just makes the conic represent an ellipse, rather than a circle.
The next condition to be satisfied is condition (2), which states that for the conic to be a circle,
Substituting for
And,
Thus, for the second condition to be satisfied, the only possible value of
Hence,
Correct answer is option (b)
Note: The general equation listed in this sum is an equation that is true for all conic sections. Imposing restrictions on this general form gives us different types of conics, like circles, ellipses, parabolas, etc. Just like we solved for the equation to be a circle right now, it can also be made to represent a parabola or any other conic simply by tweaking the variables.
Another way of approaching this problem would be to simply compare the equation given to the general form of a circle which is :
As we can see here, there’s no term containing both
Recently Updated Pages
Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Write the following in Roman numerals 25819 class 7 maths CBSE

Trending doubts
What are the public facilities provided by the government? Also explain each facility

Difference between mass and weight class 10 physics CBSE

SI unit of electrical energy is A Joule B Kilowatt class 10 physics CBSE

Statistics in singular sense includes A Collection class 10 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

The capital of British India was transferred from Calcutta class 10 social science CBSE
