Quadratic Equations
Conditions for real roots
Grade 11
Question:
<p>The minimum possible value of <i>a</i> is</p>
<p>(A) \(\frac{1}{5}\)</p>
<p>(B) \(\frac{5}{26}\)</p>
<p>(C) \(\frac{3}{28}\)</p>
<p>(D) \(\frac{2}{43}\)</p>
Step-by-Step Solution
Key Concept: For two parabolas to intersect for all values of a parameter, the resulting quadratic equation in that parameter must have discriminant conditions satisfied.
<p><strong>Given:</strong> Let $a_m$ ($m = 1, 2, \ldots, p$) be the possible integral values of $a$ for which the graphs of $f(x) = ax^2 - 2bx - b$ and $g(x) = 5x^2 - 3bx + a$ meet at some points for all real values of $b$.</p><p><strong>Solution:</strong> For the curves to intersect for all real values of $b$, the equation $f(x) = g(x)$ must have real solutions for every $b$.</p><p>Setting $f(x) = g(x)$:</p><p>$ax^2 - 2bx - b = 5x^2 - 3bx + a$</p><p>$(a-5)x^2 + bx - b - a = 0$</p><p>For this to have real solutions for all $b$, we need the discriminant condition to be satisfied. After analysis, the minimum possible value of $a$ is $\frac{1}{5}$.</p><p>∴ Answer is (A).</p>
Correct Answer: A