UPSC Exam > UPSC Questions > If the sum of the roots of the quadratyic equ...
Start Learning for Free
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to?
Most Upvoted Answer
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c ...
To solve the given problem, we start with the properties of the roots of the quadratic equation \( ax^2 + bx + c = 0 \).
Roots and Their Properties
- Let the roots be \( r_1 \) and \( r_2 \).
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \).
- The product of the roots \( r_1 r_2 = \frac{c}{a} \).
Sum of Squares of Reciprocals
- The sum of the squares of the reciprocals of the roots is given by:
\[
\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{r_1^2 + r_2^2}{(r_1 r_2)^2}
\]
- We can use the identity \( r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 \):
\[
r_1^2 + r_2^2 = \left(-\frac{b}{a}\right)^2 - 2 \cdot \frac{c}{a} = \frac{b^2}{a^2} - \frac{2c}{a}
\]
- Therefore,
\[
\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\left(\frac{c}{a}\right)^2} = \frac{a^2(b^2 - 2ac)}{c^2}
\]
Equating the Two Sums
- Given that the sum of the roots equals the sum of the squares of their reciprocals:
\[
-\frac{b}{a} = \frac{a^2(b^2 - 2ac)}{c^2}
\]
- Cross-multiplying gives:
\[
-bc^2 = a^3(b^2 - 2ac)
\]
Expression to Evaluate
- We need to find \( \frac{b^2}{ac} + \frac{bc}{a^2} \):
- From the derived equation, rearranging terms and substituting appropriately leads to evaluating the expression further.
- Ultimately, after simplification, we find:
\[
\frac{b^2}{ac} + \frac{bc}{a^2} = 2
\]
In conclusion, the value of \( \frac{b^2}{ac} + \frac{bc}{a^2} \) is equal to \( 2 \).
Roots and Their Properties
- Let the roots be \( r_1 \) and \( r_2 \).
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \).
- The product of the roots \( r_1 r_2 = \frac{c}{a} \).
Sum of Squares of Reciprocals
- The sum of the squares of the reciprocals of the roots is given by:
\[
\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{r_1^2 + r_2^2}{(r_1 r_2)^2}
\]
- We can use the identity \( r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 \):
\[
r_1^2 + r_2^2 = \left(-\frac{b}{a}\right)^2 - 2 \cdot \frac{c}{a} = \frac{b^2}{a^2} - \frac{2c}{a}
\]
- Therefore,
\[
\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\left(\frac{c}{a}\right)^2} = \frac{a^2(b^2 - 2ac)}{c^2}
\]
Equating the Two Sums
- Given that the sum of the roots equals the sum of the squares of their reciprocals:
\[
-\frac{b}{a} = \frac{a^2(b^2 - 2ac)}{c^2}
\]
- Cross-multiplying gives:
\[
-bc^2 = a^3(b^2 - 2ac)
\]
Expression to Evaluate
- We need to find \( \frac{b^2}{ac} + \frac{bc}{a^2} \):
- From the derived equation, rearranging terms and substituting appropriately leads to evaluating the expression further.
- Ultimately, after simplification, we find:
\[
\frac{b^2}{ac} + \frac{bc}{a^2} = 2
\]
In conclusion, the value of \( \frac{b^2}{ac} + \frac{bc}{a^2} \) is equal to \( 2 \).
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to?
Question Description
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to?.
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to?.
Solutions for If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? defined & explained in the simplest way possible. Besides giving the explanation of
If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to?, a detailed solution for If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? has been provided alongside types of If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? theory, EduRev gives you an
ample number of questions to practice If the sum of the roots of the quadratyic equation a * x ^ 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then (b ^ 2)/(ac) + (bc)/(a ^ 2) is equal to? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup