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If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,prove that a/b=c/d?
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If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,p...
Understanding the Problem
The given quadratic equation is:
(a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0.
We need to prove that if the roots are equal, then a/b = c/d.
Condition for Equal Roots
For a quadratic equation of the form Ax² + Bx + C = 0, the condition for equal roots is given by the discriminant being zero:
D = B² - 4AC = 0.
Here, we identify:
- A = a² + b²
- B = -2(ac + bd)
- C = c² + d²
Calculating the Discriminant
Now, we calculate the discriminant (D):
D = (-2(ac + bd))² - 4(a² + b²)(c² + d²)
Simplifying the discriminant:
D = 4(ac + bd)² - 4(a² + b²)(c² + d²)
Setting D = 0 gives us:
(ac + bd)² = (a² + b²)(c² + d²).
Applying Cauchy-Schwarz Inequality
From the equality condition of the Cauchy-Schwarz inequality:
(ac + bd)² ≤ (a² + b²)(c² + d²)
This implies that equality holds if and only if a/b = c/d.
Conclusion
Thus, we conclude that if the roots of the quadratic equation are equal, it necessarily leads to the relationship:
a/b = c/d.
This completes the proof.
The given quadratic equation is:
(a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0.
We need to prove that if the roots are equal, then a/b = c/d.
Condition for Equal Roots
For a quadratic equation of the form Ax² + Bx + C = 0, the condition for equal roots is given by the discriminant being zero:
D = B² - 4AC = 0.
Here, we identify:
- A = a² + b²
- B = -2(ac + bd)
- C = c² + d²
Calculating the Discriminant
Now, we calculate the discriminant (D):
D = (-2(ac + bd))² - 4(a² + b²)(c² + d²)
Simplifying the discriminant:
D = 4(ac + bd)² - 4(a² + b²)(c² + d²)
Setting D = 0 gives us:
(ac + bd)² = (a² + b²)(c² + d²).
Applying Cauchy-Schwarz Inequality
From the equality condition of the Cauchy-Schwarz inequality:
(ac + bd)² ≤ (a² + b²)(c² + d²)
This implies that equality holds if and only if a/b = c/d.
Conclusion
Thus, we conclude that if the roots of the quadratic equation are equal, it necessarily leads to the relationship:
a/b = c/d.
This completes the proof.
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If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,prove that a/b=c/d?
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If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,prove that a/b=c/d? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,prove that a/b=c/d? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal,prove that a/b=c/d?.
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