In a triangle, sum of lengths of two sides is $x$ and the product of length of same two sides is $y$.If${x^2} - {y^2} = {c^2}$ where $c$ the length of third side of triangle then circumradius of triangle is :
$
A)\frac{y}{{13}} \\
B)\frac{c}{{\sqrt 3 }} \\
C)\frac{c}{3} \\
D)\frac{3}{2}y \\
$
Last updated date: 18th Mar 2023
•
Total views: 306.6k
•
Views today: 8.85k
Answer
306.6k+ views
Let us consider a triangle$\Delta abc$, where $a,b,c$ are the sides of triangle
Given $c$ is the length of the third side of the triangle, then $a,b$ be the other two sides of the triangle.
Given sum of lengths of two sides = $x$
$a + b = x \to (1)$
And also given product of length of same two sides = $y$
$ab = y \to (2)$
Given condition
$ \Rightarrow $${x^2} - {y^2} = {c^2}$
Let us substitute the $x$ and $y$ values in the above equation
$
\Rightarrow {(a + b)^2} - {c^2} = ab \\
\Rightarrow {a^2} + {b^2} + 2ab - {c^2} = ab \\
\Rightarrow {a^2} + {b^2} - {c^2} = ab - 2ab \\
\Rightarrow {a^2} + {b^2} - {c^2} = - ab \\
$
Let us divide with $2ab$on both sides we get
$
\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - ab}}{{2ab}} \\
\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - 1}}{2} \to (3) \\
$
Apply the cosine rule formula where $COSC = \dfrac{{{a^2} + {b^2} - c2}}{{2ab}}$
From the cosine rule we can rewrite the equation $'3'as$
$
COSC = \dfrac{{ - 1}}{2} \\
\angle C = \dfrac{{2\pi }}{3} \\
$
We know that circumradius of triangle is $R = \dfrac{C}{{2SINC}}$
On substituting the value we get $R = \dfrac{C}{{\sqrt 3 }}$
Therefore circumradius of triangle is $R = \dfrac{C}{{\sqrt 3 }}$
( B) is the correct option
NOTE: Make a note that after substituting the value in the given condition we have divided the equation with $2ab$ on both sides, where we directly get the required value of circumradius after applying cosine rule.
Given $c$ is the length of the third side of the triangle, then $a,b$ be the other two sides of the triangle.
Given sum of lengths of two sides = $x$
$a + b = x \to (1)$
And also given product of length of same two sides = $y$
$ab = y \to (2)$
Given condition
$ \Rightarrow $${x^2} - {y^2} = {c^2}$
Let us substitute the $x$ and $y$ values in the above equation
$
\Rightarrow {(a + b)^2} - {c^2} = ab \\
\Rightarrow {a^2} + {b^2} + 2ab - {c^2} = ab \\
\Rightarrow {a^2} + {b^2} - {c^2} = ab - 2ab \\
\Rightarrow {a^2} + {b^2} - {c^2} = - ab \\
$
Let us divide with $2ab$on both sides we get
$
\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - ab}}{{2ab}} \\
\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - 1}}{2} \to (3) \\
$
Apply the cosine rule formula where $COSC = \dfrac{{{a^2} + {b^2} - c2}}{{2ab}}$
From the cosine rule we can rewrite the equation $'3'as$
$
COSC = \dfrac{{ - 1}}{2} \\
\angle C = \dfrac{{2\pi }}{3} \\
$
We know that circumradius of triangle is $R = \dfrac{C}{{2SINC}}$
On substituting the value we get $R = \dfrac{C}{{\sqrt 3 }}$
Therefore circumradius of triangle is $R = \dfrac{C}{{\sqrt 3 }}$
( B) is the correct option
NOTE: Make a note that after substituting the value in the given condition we have divided the equation with $2ab$ on both sides, where we directly get the required value of circumradius after applying cosine rule.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
