Question

If the roots of the equation q²x² + p²x + r² = 0 are the squares of the roots of the equation qx² + px + r = 0, then q, p, r are in __________

Options

• A.P.

•  G.P.

• Both A.P. and G.P.

• none of these

Answer 1

G.P.

Explanation;

Hint:

q²x² + p²x + r² = 0

Let the roots be α² + β²

α² + β² = `(−"p"^2)/("q"^2)`

α²β² = `("r"^2)/("q"^2)`

qx² + px + r = 0

Let the root be α and β

α + β = `(-"p")/("q")`, αβ = `"r"/"q"`

α² + β² = (α + β)² – 2αβ

`(-"p"^2)/("q"^2) = "p"^2/"q"^2 - (2"r")/"q"`

`(-"p"^2)/("q"^2) - ("p"^2)/("q"^2) = - (2"r")/"q"`

`(-"p"^2 - "p"^2)/("q"^2) = - (2"r")/"q"`

`(- 2"p"^2)/"q"^2 = - (2"r")/"q"`

p²q = q²r

p² = qr

`"p"/"q" = "r"/"p"`

∴ q, p, r, in G.P.