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Assertion (A): If the roots of (a^(2)+b^...

Assertion (A): If the roots of `(a^(2)+b^(2))x^(2)-2b(a+c)x+ (b^(2)+c^(2))=0` are real and equal then a, b, c are in G.P.
Reason (R): If the sum of two non-negative reals is zero then each of them is zero.

A

Both A, R are true and R explain Assertion

B

Both A, R are true but R does't explain A

C

A is true R is false

D

A is false R is true

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given quadratic equation and determine the conditions under which its roots are real and equal. The quadratic equation is: \[ (a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0 \] ### Step 1: Identify the coefficients In a standard quadratic equation of the form \(Ax^2 + Bx + C = 0\), we identify: - \(A = a^2 + b^2\) - \(B = -2b(a + c)\) - \(C = b^2 + c^2\) ### Step 2: Set the discriminant to zero For the roots to be real and equal, the discriminant \(D\) must be zero. The discriminant is given by: \[ D = B^2 - 4AC \] Substituting the values of \(A\), \(B\), and \(C\): \[ D = [-2b(a + c)]^2 - 4(a^2 + b^2)(b^2 + c^2) \] ### Step 3: Simplify the discriminant Calculating \(D\): \[ D = 4b^2(a + c)^2 - 4(a^2 + b^2)(b^2 + c^2) \] Factoring out the 4: \[ D = 4\left[b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)\right] \] ### Step 4: Set the discriminant equal to zero For the roots to be real and equal, we set \(D = 0\): \[ b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2) = 0 \] ### Step 5: Expand and rearrange the equation Expanding the left-hand side: \[ b^2(a^2 + 2ac + c^2) - (a^2b^2 + b^4 + b^2c^2) = 0 \] This simplifies to: \[ b^2a^2 + 2abc^2 + b^2c^2 - a^2b^2 - b^4 - b^2c^2 = 0 \] ### Step 6: Combine like terms Combining the terms gives: \[ b^2a^2 - a^2b^2 + 2abc^2 - b^4 = 0 \] This simplifies to: \[ 2abc^2 - b^4 = 0 \] ### Step 7: Factor the equation Factoring out \(b^2\): \[ b^2(2ac - b^2) = 0 \] ### Step 8: Solve for conditions This gives us two cases: 1. \(b^2 = 0\) which implies \(b = 0\) 2. \(2ac - b^2 = 0\) which implies \(b^2 = 2ac\) ### Step 9: Relate to geometric progression From \(b^2 = 2ac\), we can express this as: \[ \frac{b^2}{ac} = 2 \] This indicates that \(a\), \(b\), and \(c\) are in geometric progression (G.P.) if we take \(b^2 = ac\). ### Conclusion Thus, the assertion is true: if the roots of the quadratic equation are real and equal, then \(a\), \(b\), and \(c\) are in geometric progression.
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