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Suppose a,b in R. If the equation x^(2)-...

Suppose a,b `in` R. If the equation `x^(2)-(2a+b)x+(2a^(2)+b^(2)-b+1//2)=0` has two real roots, then

A

`a=1//2,b=1`

B

`a=-2,b=-1`

C

`a=2,b=-1`

D

`a=-1//2,b=1`

Text Solution

Verified by Experts

The correct Answer is:
A
Let L(h, k)
`therefore "Equation of PQ is "hx + ky = h^(2)+k^(2)`
implies Common equation of pair of lines OP and OQ is
`(x^(2))/(a^(2))+y^(2)/(b^(2))=((hx+ky)/(h^(2)+k^(2)))^(2)` (by homogenized)
`because OP_|_OQ` implies Coefficient of `x^(2)+` coefficient of `y^(2)=0`
`implies ((1)/(a^(2))+(1)/(b^(2)))(h^(2)+k^(2))=h^(2)+k^(2)`
`therefore` Locus of point L is `x^(2)+y^(2)=(a^(2)b^(2))/(a^(2)+b^(2))`
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Knowledge Check

  • Suppose a, b, c are real numbers, and each of the equations x^(2)+2ax+b^(2)=0 and x^(2)+2bx+c^(2)=0 has two distinct real roots. Then the equation x^(2)+2cx+a^(2)=0 has - (A) Two distinct positive real roots (B) Two equal roots (C) One positive and one negative root (D) No real roots

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