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Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²?
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Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b...
Given Information:
- Roots of the quadratic equation: m, n
- Coefficients of the quadratic equation: a, b, c in a non-constant GP
- Sum of reciprocals of roots: 1/m + 1/n = 1/2
Analysis:
- Let the roots be in the form m = 1/x and n = 1/(x+r), where r is a constant
- From the sum of reciprocals of roots, we get: 1/m + 1/n = x(x+r)/x(x+r) = (2x+r)/(x^2 + xr) = 1/2
- Solving the above equation, we get: r = x
- Therefore, m = 1/x and n = 1/(2x)
Calculation:
- From the quadratic equation, we have: m + n = -b/a = -a^2/b
- Substituting the values of m and n, we get: 1/x + 1/(2x) = -a^2/b
- Simplifying the above equation, we get: 3x = -a^2/b
- Therefore, a = -3x/b
Final Calculation:
- Substituting the value of a in the sum of reciprocals equation, we get: (2x + 3x)/(9x^2) = 1/2
- Solving the above equation, we get: x = 1/3
- Therefore, m = 3, n = 6
- Hence, (m-n)^2 = (3-6)^2 = 9
- Roots of the quadratic equation: m, n
- Coefficients of the quadratic equation: a, b, c in a non-constant GP
- Sum of reciprocals of roots: 1/m + 1/n = 1/2
Analysis:
- Let the roots be in the form m = 1/x and n = 1/(x+r), where r is a constant
- From the sum of reciprocals of roots, we get: 1/m + 1/n = x(x+r)/x(x+r) = (2x+r)/(x^2 + xr) = 1/2
- Solving the above equation, we get: r = x
- Therefore, m = 1/x and n = 1/(2x)
Calculation:
- From the quadratic equation, we have: m + n = -b/a = -a^2/b
- Substituting the values of m and n, we get: 1/x + 1/(2x) = -a^2/b
- Simplifying the above equation, we get: 3x = -a^2/b
- Therefore, a = -3x/b
Final Calculation:
- Substituting the value of a in the sum of reciprocals equation, we get: (2x + 3x)/(9x^2) = 1/2
- Solving the above equation, we get: x = 1/3
- Therefore, m = 3, n = 6
- Hence, (m-n)^2 = (3-6)^2 = 9
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Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²?
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Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²?.
Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²?.
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Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²?, a detailed solution for Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? has been provided alongside types of Let m and n be the roots of the equation ax²+bx-c=0, where a≠0. If a,b and c are the consecutive terms of a non constant GP and 1/m+ 1/n =1/2, then find the value of (m-n)²? theory, EduRev gives you an
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