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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 Geometric Progression 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...
Solution:
Given:
Let the roots of the equation be m and n.
a, b, and c are consecutive terms of a non-constant Geometric Progression.
1/m + 1/n = 1/2.
To Find:
Value of (m-n)².
Key Points:
- Since a, b, and c are consecutive terms of a Geometric Progression, we can write:
b² = ac
- Given that 1/m + 1/n = 1/2, we can find the value of m and n:
1/m + 1/n = (m + n) / mn = 1/2
2(m+n) = mn
- We know that the sum of the roots of a quadratic equation is given by:
m + n = -b/a
- We can substitute the value of m+n from the given equation:
2(-b/a) = mn
-2b = amn
- We know that b² = ac from the Geometric Progression property, substitute this in -2b = amn:
-2b = ab
a = -2
- Substitute the value of a in the equation -2b = ab:
-2b = -2b
b = b
- Since a, b, and c are in Geometric Progression, we have:
b² = ac
b² = -2c
- Since b is a non-zero value, c = -b²/2
- Now, we can find the roots m and n using the equations:
m + n = -b/a
2(m+n) = mn
- Substitute the value of a, b, and c in the above equations and solve for m and n:
m = 2, n = -2
- Find the value of (m-n)²:
(m-n)² = (2-(-2))² = 4² = 16
Therefore, the value of (m-n)² is 16.
Given:
Let the roots of the equation be m and n.
a, b, and c are consecutive terms of a non-constant Geometric Progression.
1/m + 1/n = 1/2.
To Find:
Value of (m-n)².
Key Points:
- Since a, b, and c are consecutive terms of a Geometric Progression, we can write:
b² = ac
- Given that 1/m + 1/n = 1/2, we can find the value of m and n:
1/m + 1/n = (m + n) / mn = 1/2
2(m+n) = mn
- We know that the sum of the roots of a quadratic equation is given by:
m + n = -b/a
- We can substitute the value of m+n from the given equation:
2(-b/a) = mn
-2b = amn
- We know that b² = ac from the Geometric Progression property, substitute this in -2b = amn:
-2b = ab
a = -2
- Substitute the value of a in the equation -2b = ab:
-2b = -2b
b = b
- Since a, b, and c are in Geometric Progression, we have:
b² = ac
b² = -2c
- Since b is a non-zero value, c = -b²/2
- Now, we can find the roots m and n using the equations:
m + n = -b/a
2(m+n) = mn
- Substitute the value of a, b, and c in the above equations and solve for m and n:
m = 2, n = -2
- Find the value of (m-n)²:
(m-n)² = (2-(-2))² = 4² = 16
Therefore, the value of (m-n)² is 16.
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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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression 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 Geometric Progression and 1/m+ 1/n =1/2, then find the value of (m-n)²? theory, EduRev gives you an
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