If 2 and 6 are the roots of the equation a x²+b x+1=0, then the quadratic equation, whose roots are 1/(2 a+b) and 1/(6 a+b), is :

Solution

Maths sir · 4 steps

Use the given roots to find a and b first. After that, the new roots are just two numbers to work out.

1. Use the roots. For ax² + bx + 1 = 0, the product of roots is 1/a and the sum is -b/a. Here the product is 2 × 6 = 12 and the sum is 2 + 6 = 8.

   1/a = 12 ⇒ a = 1/12, -b/a = 8 ⇒ b = -8/12 = -2/3

2. Find new roots. 2a + b = 1/6 - 2/3 = -1/2, so the first root is 1/(2a+b) = -2. Also 6a + b = 1/2 - 2/3 = -1/6, so the second root is 1/(6a+b) = -6.
3. Build the equation. The new roots have sum -8 and product 12. So the equation is x² - (-8)x + 12 = 0, which is x² + 8x + 12 = 0.
4. So. The answer is (a). Check: (x+2)(x+6) = x² + 8x + 12.

The idea to keep. When a question gives you the roots, Vieta's relations turn them straight into the coefficients. Find those first, and the rest is plain arithmetic.

Answer from the NTA final answer key. Solution by PrepWiser.