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The equation whose roots are p−qp+q\frac{p - q}{p + q}p+qp−q​​ and −p+qp−q-\frac{p + q}{p - q}−p−qp+q​​ will be:
Question

The equation whose roots are p−qp+q\frac{p - q}{p + q}​ and −p+qp−q-\frac{p + q}{p - q}​ will be:

A.

​(p2+q2)x2+8pqx−2(p+q)2=0(p^2 + q^2)x^2 + 8pqx - 2(p + q)^2 = 0​​

B.

​2pq2x2+pqx+2(p+q)2=0\frac{2pq^2}{x^2} + pqx + 2(p + q)^2 = 0​​

C.

​(p2−q2)x2+4pqx−p2+q2=0(p^2 - q^2)x^2 + 4pqx - p^2 + q^2 = 0​​

D.

​p2q2x2+pqx+(p+q)2=0\frac{p^2}{q^2}x^2 + \frac{p}{q}x + (p + q)^2 = 0 ​​

Correct option is C

Given: Roots are
​α=(p−q)/(p+q),β=−(p+q)/(p−q)\alpha = (p - q)/(p + q),\beta = -(p + q)/(p - q)​​
Concept used
If roots of a quadratic equation are alpha and beta, the equation is:
​x2−(α+β)x+α×β=0x^2 - (\alpha + \beta)x + \alpha\times \beta = 0​​
 Sum of roots
​α+β=(p−q)/(p+q)−(p+q)/(p−q)=((p−q)2−(p+q)2)/((p+q)(p−q))=(p2−2pq+q2−p2−2pq−q2)/(p2−q2)=−4pq/(p2−q2)\alpha + \beta = (p - q)/(p + q) - (p + q)/(p - q)\\= ((p - q)^2 - (p + q)^2) / ((p + q)(p - q))\\= (p^2 - 2pq + q^2 - p^2 - 2pq - q^2) / (p^2 - q^2)\\= -4pq / (p^2 - q^2)​​
Product of roots
​α∗β=[(p−q)/(p+q)]∗[−(p+q)/(p−q)]=−1\alpha * \beta = [(p - q)/(p + q)] * [-(p + q)/(p - q)] = -1​​
Form the equation
​x2+(4pq/(p2−q2))x−1=0x^2 + (4pq / (p^2 - q^2))x - 1 = 0​​

Multiply through by (p2−q2): (p^2 - q^2):​​
​(p2−q2)x2+4pqx−(p2−q2)=0(p^2 - q^2)x^2 + 4pqx - (p^2 - q^2) = 0​​
Correct answer is (C) (p2−q2)x2+4pqx−p2+q2=0(p^2 - q^2)x^2 + 4pqx - p^2 + q^2 = 0​​

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