Answer
Verified
412.8k+ views
Hint: In this question, we first need to find the length of the arc using the formula \[l=r\theta \]. Then substitute this value of arc length in the area of the sector formula given by \[A=\dfrac{1}{2}rl\] and simplify further to get the result.
Complete step-by-step answer:
Now, form the given conditions in the question we have angle subtended at the centre as 90 degrees which gives
\[\theta ={{90}^{\circ }}\]
Now, this can also be written in the radians as
\[\theta =\dfrac{\pi }{2}\]
Now, as we already know that in the sector of a circle the length of the arc is given by the formula
\[l=r\theta \]
Here, l is the length of the arc, r is the radius of the corresponding circle and \[\theta \]is the angle subtended by the arc at the centre
Now, let us assume that the radius of the circle as r and length of the arc as l
Given in the question that \[\theta =\dfrac{\pi }{2}\]
Let us draw the diagram with the above given conditions
Now, from the length of the arc formula on substituting the respective values we get,
\[\Rightarrow l=r\theta \]
Now, on further substitution we get,
\[\Rightarrow l=\dfrac{\pi r}{2}\]
Let us also assume that the area of the circle as A
Now, we can find the area of the sector using the formula
\[A=\dfrac{1}{2}rl\]
As we already have the value of length of arc from above calculations we get,
\[l=\dfrac{\pi r}{2}\]
Now, on substituting the respective values in the area of sector formula we get,
\[\Rightarrow A=\dfrac{1}{2}\times r\times \dfrac{\pi r}{2}\]
Now, on simplifying it further we get,
\[\therefore A=\dfrac{\pi {{r}^{2}}}{4}\]
Hence, the correct option is (c).
Note:Instead of converting the given angle to radians we can also use the respective formula in terms of degrees and simplify further to get the answer. Both the methods give the same result.It is important to note that finding the length of the arc is not necessary because we can substitute the length formula in the area formula and then directly substitute the angle to get the result.
Complete step-by-step answer:
Now, form the given conditions in the question we have angle subtended at the centre as 90 degrees which gives
\[\theta ={{90}^{\circ }}\]
Now, this can also be written in the radians as
\[\theta =\dfrac{\pi }{2}\]
Now, as we already know that in the sector of a circle the length of the arc is given by the formula
\[l=r\theta \]
Here, l is the length of the arc, r is the radius of the corresponding circle and \[\theta \]is the angle subtended by the arc at the centre
Now, let us assume that the radius of the circle as r and length of the arc as l
Given in the question that \[\theta =\dfrac{\pi }{2}\]
Let us draw the diagram with the above given conditions
Now, from the length of the arc formula on substituting the respective values we get,
\[\Rightarrow l=r\theta \]
Now, on further substitution we get,
\[\Rightarrow l=\dfrac{\pi r}{2}\]
Let us also assume that the area of the circle as A
Now, we can find the area of the sector using the formula
\[A=\dfrac{1}{2}rl\]
As we already have the value of length of arc from above calculations we get,
\[l=\dfrac{\pi r}{2}\]
Now, on substituting the respective values in the area of sector formula we get,
\[\Rightarrow A=\dfrac{1}{2}\times r\times \dfrac{\pi r}{2}\]
Now, on simplifying it further we get,
\[\therefore A=\dfrac{\pi {{r}^{2}}}{4}\]
Hence, the correct option is (c).
Note:Instead of converting the given angle to radians we can also use the respective formula in terms of degrees and simplify further to get the answer. Both the methods give the same result.It is important to note that finding the length of the arc is not necessary because we can substitute the length formula in the area formula and then directly substitute the angle to get the result.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Net gain of ATP in glycolysis a 6 b 2 c 4 d 8 class 11 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE