An area situation gives l² - 2(a + 3)l + (a² + 10) = 0. What is the condition on a for real length values?
Answer and explanation
Correct answer: a ≥ 1/6
A real value of l can occur only when the quadratic equation has real roots, so its discriminant must satisfy D ≥ 0. Comparing l² - 2(a + 3)l + (a² + 10) with Al² + Bl + C, we have A = 1, B = -2(a + 3), and C = a² + 10. Thus D = B² - 4AC = 4(a + 3)² - 4(a² + 10) = 4[(a² + 6a + 9) - a² - 10] = 4(6a - 1). The condition 4(6a - 1) ≥ 0 gives a ≥ 1/6. Therefore option A is correct. The other choices either reverse the inequality, impose an unnecessary single value, or ignore the discriminant condition.
Frequently asked questions
What is the correct answer to this question?
a ≥ 1/6
Why is this the correct answer?
A real value of l can occur only when the quadratic equation has real roots, so its discriminant must satisfy D ≥ 0. Comparing l² - 2(a + 3)l + (a² + 10) with Al² + Bl + C, we have A = 1, B = -2(a + 3), and C = a² + 10. Thus D = B² - 4AC = 4(a + 3)² - 4(a² + 10) = 4[(a² + 6a + 9) - a² - 10] = 4(6a - 1). The condition 4(6a - 1) ≥ 0 gives a ≥ 1/6. Therefore option A is correct. The other choices either reverse the inequality, impose an unnecessary single value, or ignore the discriminant condition.
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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