Assertion: For the equation \(x^2-2(a+b)x+(a-b)^2=0\), if \(ab>0\), its roots are real and distinct. Reason: The discriminant of this equation is \(D=16ab\). Choose the correct option.
Answer and explanation
Correct answer: Both the assertion and the reason are correct, and the reason correctly explains the assertion
Here, \(A=1\), \(B=-2(a+b)\), and \(C=(a-b)^2\). Thus, \(D=B^2-4AC=4(a+b)^2-4(a-b)^2=16ab\). Since \(ab>0\), we have \(D>0\), so the roots are real and distinct. Therefore, the reason is correct and directly explains the assertion. Exam tip: Determine the sign of \(D\) first; \(D>0\) indicates two real and distinct roots.
Frequently asked questions
What is the correct answer to this question?
Both the assertion and the reason are correct, and the reason correctly explains the assertion
Why is this the correct answer?
Here, \(A=1\), \(B=-2(a+b)\), and \(C=(a-b)^2\). Thus, \(D=B^2-4AC=4(a+b)^2-4(a-b)^2=16ab\). Since \(ab>0\), we have \(D>0\), so the roots are real and distinct. Therefore, the reason is correct and directly explains the assertion. Exam tip: Determine the sign of \(D\) first; \(D>0\) indicates two real and distinct roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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