Question

The complete set of values of 'a' for which the equation 9^x -(4a)3^x + 4 - a² = 0 has a unique root in the interval (0,1).

Answer 1

\(9^x - (4a)3^x + 4 - a^2 = 0\)

\(D = 0\)

\(3^x = y\)

\(y^2 - 4ay + (4 - a^2) = 0\)

\(\Rightarrow b^2 - 4ac = D\)

\(\Rightarrow 16a^2 - 4(4 - a^2) = 0\)

\(\Rightarrow 16a^2 - 16+ 4a^2 = 0\)

\(\Rightarrow 20a^2 = 16\)

\(\Rightarrow a = \pm \frac 2{\sqrt 5}\)

\(\therefore 3^x = \frac{-(-4a)}{2(1)} = \frac{4a}2 = 2a\)

⇒ \(x = \log_3 (2a)\)

\(\therefore a = \frac 2{\sqrt 5}\)