Home
Class 12
MATHS
If the roots of the equation 1/ (x+p) + ...

If the roots of the equation `1/ (x+p) + 1/ (x+q) = 1/r` are equal in magnitude but opposite in sign, show that `p+q = 2r` & that the product of roots is equal to `(-1/2)(p^2+q^2)`.

Text Solution

Verified by Experts

We have `1/(x+p)+1/(x+q)=1/r`
`implies((x+q)+(x+p))/(x^(2)+(p+q)x+pq)=1/r`
`impliesx^(2)+(p+q-2r)x+pq-(p+q)r=0`
Now since the roots are equal in magnitudes but opposite in sign. Therefore
Sum of the roots `=0`
`impliesp+q-2r=0`
`impliesp+q=2r`....i
and product fo the roots `=pq-(p+q)r`
`=pq-(p+q)((p+q)/2)` [from Eq (i) ]
`=(2pq-p^(2)-q^(2)-2pq)/2`
`=-(p^(2)+q^(2))/2`
Promotional Banner

Similar Questions

Explore conceptually related problems

The value of m for which the equation x^(3) + x + 1 = 0 . Has two roots equal in magnitude but opposite in sign, is :

The roots of the equation (q- r) x^(2) + (r - p) x + (p - q)= 0 are

If p and q are the roots of the equation x^2-p x+q=0 , then

For what value of m will the equation (x^2-bx)/(ax-c)=(m-1)/(m+1) have roots equal in magnitude but opposite in sign?

If p and q are the roots of the equation x^(2)+p x+q=0 then

If the product of the roots of the equations x^(2) + 3x + q=0 is zero then q is equal to :

If p and q are the roots of the equations x^(2) - 3x+ 2 = 0 , find the value of (1)/(p) - (1)/(q)

If 2+i sqrt(3) is a root of the equation x^(2)+p x+q=0 , where p, q are real, then (p, q)=

The set of values of p for which the roots of the equation 3 x^(2)+2 x+p(p-1)=0 are of opposite signs is

If one root of the equation x^(2)+p x+12=0 is 4, while the equation x^(2)+p x+q=0 has equal roots then the value of q is