Question

If α and β are the roots of the equation (a − 2) x^2 − (5 − a) x − 5 = 0. Find a if |α - β| = 2√6.​

Answer 1

Answer:

So the required answer is √[(25+a²-10a)/(a²+4-4a) +20/(a-2)].

Explanation:

We know that for equation ax²+bx+c=0,

Sum of roots = -b/a

Product of roots = c/a

Now , for given equation (a − 2) x^2 − (5 − a) x − 5 = 0.

roots are α and β.

So, α + β = (5-a)/(a-2). (1)

α.β = -5/(a-2)

On squaring equation (1), we get

(α+β)² = {(5-a)/(a-2)}²

⇔α²+β²+2αβ = (25+a²-10a)/(a²+4-4a).

⇔α²+β²+2.{(-5)/a-2} = (25+a²-10a)/(a²+4-4a)

⇔α²+β²=(25+a²-10a)/(a²+4-4a) - 2.{(-5)/a-2}

Also, (α-β)² = α²+β²-2αβ = (25+a²-10a)/(a²+4-4a) - 2.{(-5)/a-2}- 2.{(-5)/a-2}.

⇔|α-β| = √[(25+a²-10a)/(a²+4-4a) - 2.{(-5)/a-2}- 2.{(-5)/a-2}]

=✓[(25+a²-10a)/(a²+4-4a) - 4.{(-5)/a-2}].

=√[(25+a²-10a)/(a²+4-4a) +20/(a-2)].