If the arithmetic mean and the geometric mean of two numbers a and b are denoted by A and G respectively, then form a quadratic equation whose roots are a and b.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(x^2-2Ax+G^2=0\)
A quadratic equation whose roots are \(a\) and \(b\) is \(x^2-(\text{sum of roots})\,x+(\text{product of roots})=0\). The arithmetic mean gives the sum of the two numbers and the geometric mean gives their product, so both coefficients come straight from the means.
The arithmetic mean is \(\displaystyle A=\frac{a+b}{2}\), so the sum of the roots is \(a+b=2A\)
The geometric mean is \(G=\sqrt{ab}\), so the product of the roots is \(ab=G^2\)
Putting the sum and the product into the form: \(x^2-2Ax+G^2=0\)