Question
If $a + b + c = 0$, then write the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}.$
✓
Answer
We have to find the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}$
Given $a + b + c = 0$
Using identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
Put $a + b + c = 0 a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - bc - ca) a^3 + b^3 + c^3 - 3$
$abc$ $= 0 a^3 + b^3 + c^3 = 3abc$
$\frac{\text{a}^3}{\text{abc}}+\frac{\text{b}^3}{\text{abc}}+\frac{\text{c}^3}{\text{abc}}=3$
$\frac{\text{a}\times\text{a}\times\text{a}}{\text{abc}}+\frac{\text{b}\times\text{b}\times\text{b}}{\text{abc}}+\frac{\text{c}\times\text{c}\times\text{c}}{\text{abc}}=3$
$\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}=3$
Hence the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}$ is $3$.
Given $a + b + c = 0$
Using identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
Put $a + b + c = 0 a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - bc - ca) a^3 + b^3 + c^3 - 3$
$abc$ $= 0 a^3 + b^3 + c^3 = 3abc$
$\frac{\text{a}^3}{\text{abc}}+\frac{\text{b}^3}{\text{abc}}+\frac{\text{c}^3}{\text{abc}}=3$
$\frac{\text{a}\times\text{a}\times\text{a}}{\text{abc}}+\frac{\text{b}\times\text{b}\times\text{b}}{\text{abc}}+\frac{\text{c}\times\text{c}\times\text{c}}{\text{abc}}=3$
$\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}=3$
Hence the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}$ is $3$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Show that $1.272727 = 1. \overline {27}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0.$
→In figure if $l\ ||\ m, n∥p$ and $\angle\text{1}=85^\circ,$ find $\angle{2}.$
→The radius of the internal and external surfaces of a hollow spherical shell are $3\ cm$ and $5\ cm$ respectively. If it is melted and recast into a solid cylinder of height $2\frac{2}{3}$cm. Find the diameter of the cylinder.
→In fig lines $XY$ and $MN$ intersect at $O$ If $\angle POY = 90$^\circ and $a:b = 2:3$ find $\angle c.$
→A dome of a building is in the form of a hemisphere. From inside, it was white$-$washed at the cost of $Rs. 4989.60$. If the cost of white$-$washing is $Rs. 20$ per square metre. Find the
$i.$ inside surface area of the dome.
$ii.$ Volume of the air inside the dome.
→$i.$ inside surface area of the dome.
$ii.$ Volume of the air inside the dome.
Insert $16$ rational numbers between $2.1$ and $2.2$
→If $x-2$ is a factor of the following two polynomials, find the values of a in case:
$x^3-2 a x^2+a x-1$
→$x^3-2 a x^2+a x-1$
The perimeter of a parallelogram is $22\ cm$.If the longer side measures $6.5\ cm$ what is the measure of the shorter side?
→Find the sum of the deviations of the variate values $3, 4, 6, 7, 8, 14$ from their mean.
→If $x = 3$ and $y = -1$, find the values of the following using in identity: $\Big(\frac{\text{x}}{4}-\frac{\text{y}}{3}\Big)\Big(\frac{\text{x}^2}{16}+\frac{\text{xy}}{12}+\frac{\text{y}^2}{9}\Big)$
→