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If the roots of the equation (a2 + b2) x2 - 2b(a + c) x + (b2 + c2) = 0 are equal then a, b, c, are in
- a)AP
- b)GP
- c)HP
- d)Cannot be said
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the roots of the equation (a2 + b2) x2 - 2b(a + c) x + (b2 + c2) = ...
Solve by assuming values of a, b, and c in AP, GP and HP to check which satisfies the condition.
Most Upvoted Answer
If the roots of the equation (a2 + b2) x2 - 2b(a + c) x + (b2 + c2) = ...
Explanation:
To find the relation between a, b, and c when the roots of the given equation are equal, let's analyze the equation step by step.
Given equation: (a^2 - b^2)x^2 - 2b(a - c)x + (b^2 - c^2) = 0
Step 1:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
Step 2:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the expression b^2 - 4ac.
Step 3:
In the given equation, the discriminant is: (-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
Step 4:
To simplify the expression, let's expand the terms and then simplify:
(-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a - c)^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a^2 - 2ac + c^2) - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2a^2 - 8b^2ac + 4b^2c^2 - 4a^2b^2 + 8ac^2 - 4b^2c^2
= 4b^2a^2 - 4a^2b^2 + 8ac^2 - 8b^2ac + 4b^2c^2 - 4b^2c^2
= 4(a^2b^2 - a^2b^2 + 2ac^2 - 2b^2ac)
= 4(2ac^2 - 2b^2ac)
= 8ac^2 - 8ab^2c
Step 5:
Setting the discriminant equal to zero, we have:
8ac^2 - 8ab^2c = 0
Step 6:
Factor out 8ac from the equation:
8ac(c - b^2) = 0
Step 7:
For the equation to be true, either 8ac = 0 or (c - b^2) = 0.
Step 8:
If 8ac = 0, it implies either a = 0 or c = 0.
Step 9:
If (c - b^2) = 0, it implies c = b^2.
Conclusion:
From the above analysis, we can conclude that a, b, and c are in a geometric progression (GP) when the roots of the given equation are equal. Therefore, the correct answer is option 'B' (GP).
To find the relation between a, b, and c when the roots of the given equation are equal, let's analyze the equation step by step.
Given equation: (a^2 - b^2)x^2 - 2b(a - c)x + (b^2 - c^2) = 0
Step 1:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
Step 2:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the expression b^2 - 4ac.
Step 3:
In the given equation, the discriminant is: (-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
Step 4:
To simplify the expression, let's expand the terms and then simplify:
(-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a - c)^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a^2 - 2ac + c^2) - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2a^2 - 8b^2ac + 4b^2c^2 - 4a^2b^2 + 8ac^2 - 4b^2c^2
= 4b^2a^2 - 4a^2b^2 + 8ac^2 - 8b^2ac + 4b^2c^2 - 4b^2c^2
= 4(a^2b^2 - a^2b^2 + 2ac^2 - 2b^2ac)
= 4(2ac^2 - 2b^2ac)
= 8ac^2 - 8ab^2c
Step 5:
Setting the discriminant equal to zero, we have:
8ac^2 - 8ab^2c = 0
Step 6:
Factor out 8ac from the equation:
8ac(c - b^2) = 0
Step 7:
For the equation to be true, either 8ac = 0 or (c - b^2) = 0.
Step 8:
If 8ac = 0, it implies either a = 0 or c = 0.
Step 9:
If (c - b^2) = 0, it implies c = b^2.
Conclusion:
From the above analysis, we can conclude that a, b, and c are in a geometric progression (GP) when the roots of the given equation are equal. Therefore, the correct answer is option 'B' (GP).
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