Question

If $\alpha$,$\beta$ are the roots of $x^2+px+1=0$ and $\gamma$,$\delta$ are the roots of $x^2+qx+1=0$, then $q^2-p^2= $

A) $(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)$

B) $(\alpha+\gamma)(\beta+\gamma)(\alpha-\delta)(\beta+\delta)$

C) $(\alpha+\gamma)(\beta+\gamma)(\alpha+\delta)(\beta+\delta)$

D) None of these

Answer 1

Hint: To solve this problem, we will use the properties of roots of a quadratic equation. By understanding the relationship between the coefficients and roots of a quadratic equation, we will be able to simplify the given expressions and find the correct answer.



Formula Used:We know that the roots of a quadratic equation of the form $ax^2+bx+c=0$ are given by $(-b+\sqrt{b^2-4ac})/2a$ and $(-b-\sqrt{b^2-4ac})/2a$.



Complete step by step solution:Using the above formula, we can find the roots of the first equation $x^2+px+1=0$ as $(-p+\sqrt{p^2-4})/2$ and $(-p-\sqrt{p^2-4})/2$. Let's call these roots $\alpha$ and $\beta$.

Similarly, we can find the roots of the second equation $x^2+qx+1=0$ as $(-q+\sqrt{q^2-4})/2$ and $(-q-\sqrt{q^2-4})/2$. Let's call these roots $\gamma$ and $\delta$.

Now, we will use the given equation $q^2-p^2= (\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)$ and substitute the values of $\alpha$, $\beta$, $\gamma$ and $\delta$ in this equation.

After simplifying, we will be able to compare this with the given options and find the correct answer.

Upon substituting the roots values in the given equation we get $(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta) = (\alpha-\beta)(\gamma-\delta)(\alpha+\beta)(\gamma+\delta) = (\alpha+\beta)(\gamma+\delta)^2-(\alpha-\beta)(\gamma-\delta)^2 = (p+q)^2- (p-q)^2 = q^2-p^2$



Option ‘A’ is correct



Note: The key to solving this problem is to understand the relationship between the roots and coefficients of a quadratic equation. By substituting the values and simplifying them, we will be able to find the correct answer.