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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?
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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 \( 2 \).
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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?
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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?.
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