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If the equation
(1 + t²) x² + 2 t c x + (c² – a²) = 0
has equal roots then which of the following is true?
(1 + t²) x² + 2 t c x + (c² – a²) = 0
has equal roots then which of the following is true?
- a)t² = c² (1 + a²)
- b)a² = c² (1 + t²)
- c)c² = a² (1 + t²)
- d)↵More than one of the above
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the equation(1 + t²) x² + 2 t c x + (c² – a&su...
Understanding the Problem
To determine the conditions for the quadratic equation
(1 + t²)x² + 2tcx + (c² - a²) = 0
to have equal roots, we need to analyze its discriminant.
Discriminant Condition
For a quadratic equation Ax² + Bx + C = 0, the condition for equal roots is:
B² - 4AC = 0.
In our case:
- A = 1 + t²
- B = 2tc
- C = c² - a²
We substitute these values into the discriminant condition:
(2tc)² - 4(1 + t²)(c² - a²) = 0.
This simplifies to:
4t²c² - 4(1 + t²)(c² - a²) = 0.
Rearranging the Equation
Dividing through by 4, we get:
t²c² = (1 + t²)(c² - a²).
Expanding the right-hand side results in:
t²c² = c² + t²c² - a² - t²a².
Simplifying this gives:
0 = c² - a² - t²a².
Isolating a²
Rearranging the equation provides:
a² = c²(1 + t²).
This confirms that option B is correct: a² = c²(1 + t²).
Conclusion
Thus, for the quadratic equation to have equal roots, the condition that must hold true is option B.
To determine the conditions for the quadratic equation
(1 + t²)x² + 2tcx + (c² - a²) = 0
to have equal roots, we need to analyze its discriminant.
Discriminant Condition
For a quadratic equation Ax² + Bx + C = 0, the condition for equal roots is:
B² - 4AC = 0.
In our case:
- A = 1 + t²
- B = 2tc
- C = c² - a²
We substitute these values into the discriminant condition:
(2tc)² - 4(1 + t²)(c² - a²) = 0.
This simplifies to:
4t²c² - 4(1 + t²)(c² - a²) = 0.
Rearranging the Equation
Dividing through by 4, we get:
t²c² = (1 + t²)(c² - a²).
Expanding the right-hand side results in:
t²c² = c² + t²c² - a² - t²a².
Simplifying this gives:
0 = c² - a² - t²a².
Isolating a²
Rearranging the equation provides:
a² = c²(1 + t²).
This confirms that option B is correct: a² = c²(1 + t²).
Conclusion
Thus, for the quadratic equation to have equal roots, the condition that must hold true is option B.
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Community Answer
If the equation(1 + t²) x² + 2 t c x + (c² – a&su...
In the given equation,
a = (1 + t²)
b = 2 t c
c = (c² – a²)
a = (1 + t²)
b = 2 t c
c = (c² – a²)
⇒ b² – 4 a c = 0
⇒ 4 t² c² – 4 (1 + t²) (c² – a²) = 0
⇒ t² c² – t² c² + a² – c² + a² t² = 0
⇒ a² (1 + t²) = c²
∴ a² (1 + t²) = c² is true.
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Question Description
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