Let a, b be non zero real numbers such that ab=5/2 and given A=a-bba and AAT=20I (I is a unit matrix), then the equation whose roots are a and b is

AP EAMCET · Maths · Matrices

Let $a, b$ be non zero real numbers such that $a b = 5 / 2$ and given $A = [\begin{matrix} a & - b \\ b & a \end{matrix}]$ and $A A^{T} = 20 I$ ( $I$ is a unit matrix), then the equation whose roots are $a$ and $b$ is

A $x^{2} \mp 10 x + 5 = 0$
B $2 x^{2} \pm 10 x + 5 = 0$
C $x^{2} - 5 x + \frac{5}{2} = 0$
D $x^{2} - 25 x + \frac{5}{2} = 0$

Verified Solution

Answer & Solution

Correct Answer

(B) $2 x^{2} \pm 10 x + 5 = 0$

Step-by-step Solution

Detailed explanation

Given that, A=a-bba and ab=52 ....(i) ⇒AT=ab-ba ⇒AAT=a2+b200a2+b2 Also given AAT=20I ⇒a2+b200a2+b2=200020 ⇒a2+b2=20 ⇒a+b2-2ab=20 ⇒a+b2-2×52=20 [from i] ⇒a+b2=25 ⇒a+b=±5 Now, we know that general form of a quadratic…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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