If the ratio of roots of the equation x^2 + px + q = 0 is equal to the ratio of roots of x^2 + rx + m = 0, prove that m.p^2 = q.r^2

First of all, lets have a look at what we're given. Ratio of roots of x^2 + px + q = 0 and x^2 + rx + m = 0 are equal.

We are assuming that the roots of x^2 + px + q = 0 are α, β while roots of x^2 + rx + m = 0 are γ, δ. So, as we're given,

α / β = γ / δ

By using the sum and product of roots formulae, we can say that

α + β = -p ; αβ = q
γ + δ = -r ; γδ = m

We have to prove that m.p^2 = q.r^2. Now, there can be several approaches to the proof. We'll be telling you two of them.

The ratio of roots of the equation x^2 + ax + a + 2 = 0 is 2. Find the value of a.

The equation given is x^2 + ax + a + 2 = 0 and it is told that the 'ratio' of its roots is 2. 

Now, if you think it over, you'll understand that if two numbers are in ratio 2, then one of those numbers have to be 2 times the other. Or, in other words we can say, that one of the number will be half (1/2 times) the other. Both these conditions are the same.

So, we can assume the roots of x^2 + ax + a + 2 = 0 to be t and 2t. 

(You can see it here. 2t is 2 times t, and obviously then, t is 1/2 times 2t)

So, here,

Sum of roots = -(a)/1

=> t + 2t = -a

=> 3t = -a

=> t = -a/3

Product of roots = (a + 2)/1

=> t * 2t = (a + 2)

=> 2t^2 = (a + 2)