If A.M. and G.M. of x and y are in the ratio p:q, then x:y is

Question

If A.M. and G.M. of $x$ and $y$ are in the ratio $p : q$ , then $x : y$ is

Smart Achievers · Accepted Answer

(2)

$A = \frac{x + y}{2}$ $and$ $G = \sqrt{x y}$

$x + y = 2 A$ $and$ $G^{2} = x y$

$The equation having x, y as its roots is$

$t^{2} - (x + y) t + x y = 0$ $or,$ $t^{2} - 2 A t + G^{2} = 0$

$∴$ $t = \frac{2 A \pm \sqrt{4 A^{2} - 4 G^{2}}}{2} \Rightarrow t = A \pm \sqrt{A^{2} - G^{2}}$

$So, the two numbers are$

$x = A + \sqrt{A^{2} - G^{2}}$ $and$ $y = A - \sqrt{A^{2} - G^{2}}$

$It is given that A : G = p : q \Rightarrow A = λ p$ $and$ $G = λ q$

$Now, substituting the values of A and G in$

$x = A + \sqrt{A^{2} - G^{2}}$ $and$ $y = A - \sqrt{A^{2} - G^{2}}, we get$

$\frac{x}{y} = \frac{λ p + \sqrt{λ^{2} p^{2} - λ^{2} q^{2}}}{λ p - \sqrt{λ^{2} p^{2} - λ^{2} q^{2}}} \Rightarrow \frac{x}{y} = \frac{p + \sqrt{p^{2} - q^{2}}}{p - \sqrt{p^{2} - q^{2}}}$

$∴$ $x : y = (p + \sqrt{p^{2} - q^{2}}) : (p - \sqrt{p^{2} - q^{2}})$

Solution

Q.
If A.M. and G.M. of $x$ and $y$ are in the ratio $p : q$ , then $x : y$ is

1 $p - \sqrt{p^{2} + q^{2}} : p + \sqrt{p^{2} + q^{2}}$

2 $p + \sqrt{p^{2} - q^{2}} : p - \sqrt{p^{2} - q^{2}}$

3 $p : q$

4 $p + \sqrt{p^{2} + q^{2}} : p - \sqrt{p^{2} + q^{2}}$

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Solution

Q. If A.M. and G.M. of x and y are in the ratio p:q, then x:y is

1 p-p2+q2:p+p2+q2

2 p+p2-q2:p-p2-q2

3 p:q