If x and y are non zero real numbers satisfying $x y (x^{2} - y^{2}) = x^{2} + y^{2}$ , find the minimum value of $x^{2} + y^{2}$ .

Open in App

Solution

Given, $x y (x^{2} - y^{2}) = x^{2} + y^{2}$
Let $f^{1} (x, y) = x^{2} + y^{2}$ and $f^{2} (x, y) = x y (x^{2} - y^{2})$
$x^{2} + y^{2} = x y (x^{2} - y^{2})$ .....(1)

Also, when $f^{1} (x, y) = f^{2} (x, y)$ , the slope at that point will be equal for both.
$\Rightarrow \frac{d}{d x} (x^{2} + y^{2}) = 2 x + 2 y \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{(- x)}{y}$
and $\frac{d}{d x} x y (x^{2} - y^{2}) = (3 x^{2} y + x^{3} \frac{d y}{d x} - 3 y^{2} x \frac{d y}{d x} - y^{3}) = 0 \Rightarrow \frac{d y}{d x} = \frac{y^{3} - 3 x^{2} y}{x^{3} - 3 y^{2} x} = \frac{y}{x} (\frac{y^{2} - 3 x^{2}}{x^{2} - 3 y^{2}})$

ie, $\frac{(- x)}{y} = \frac{y}{x} (\frac{y^{2} - 3 x^{2}}{x^{2} - 3 y^{2}}) \Rightarrow x^{2} (3 y^{2} - x^{2}) = y^{2} (y^{2} - 3 x^{2}) \Rightarrow x^{2} + y^{2} - 2 x^{2} y^{2} = 4 x^{2} y^{2} \Rightarrow (x^{2} - y^{2})^{2} = (2 x y)^{2}$
$\Rightarrow x^{2} - y^{2} = 2 x y$ .....(2)

NOTE: $x^{2} - y^{2} \neq - 2 x y$
( $∵$ Substituting (2) in (1) makes $x^{2} + y^{2}$ negetive)

Dividing (2) with $y^{2}$ ,
$\frac{x^{2}}{y^{2}} - 1 = 2 \frac{x}{y} \Rightarrow \frac{x^{2}}{y^{2}} - 2 \frac{x}{y} + 1 = 2 \Rightarrow (\frac{x}{y} - 1)^{2} = 2$
ie, $\frac{x}{y} = (\sqrt{2} - 1) \Rightarrow x = (\sqrt{2} - 1) y ⟶ (3)$

NOTE: $\frac{x}{y} \neq (- \sqrt{2} - 1)$
( $∵ x > 0$ and $y > 0$ )

Substituting (3) in (1),
$x y = \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{[(\sqrt{2} - 1)^{2} + 1] y^{2}}{[(\sqrt{2} - 1)^{2} - 1] y^{2}} \Rightarrow x y = \frac{4 + 2 \sqrt{2}}{2 + 2 \sqrt{2}} = \frac{2 \sqrt{2} (\sqrt{2} + 1)}{2 (\sqrt{2} + 1)}$
ie, $\Rightarrow x y = \sqrt{2} ⟶ (4)$

Substituting (4) and (2) in (1),
$x^{2} + y^{2} = x y (2 x y) = 2 (\sqrt{2})^{2} = 4$

So, in this case, $x^{2} + y^{2}$ is minimum at $4$ .

Suggest Corrections

Similar questions

Q. Let x & y be the real numbers satisfying the equation

x^{2} - 4 x + y^{2} + 3 = 0

.If the maximum and minimum values of

x^{2} + y^{2}

are M & m respectively , then find the numerical value of (M+m).

Q. If x and y are real number such that

x^{2} + 2 x y - y^{2} = 6,

find the minimum value of

{(x^{2} + y^{2})}^{2} .

Q. Let

x

and

y

be real numbers satisfying the inequality

5 x^{2} + y^{2} - 4 x y + 24 \leq 10 x - 1.

Find the value of

x^{2} + y^{2} - 29.

(correct answer + 2, wrong answer 0)

Let (x, y) be a pair of real number satisfying
$56 x + 33 y = \frac{- y}{x^{2} + y^{2}}$ and
$33 x - 56 y = \frac{x}{x^{2} + y^{2}}$ . If |x| + |y| = $\frac{p}{q}$ (where p and q are relatively prime), then (6p – q)is ___

Q. Number of real ordered pair

(x, y)

satisfying

x^{2} + 1 = y

and

y^{2} + 1 = x

Join BYJU'S Learning Program

If x and y are non zero real numbers satisfying xy(x2−y2)=x2+y2, find the minimum value of x2+y2.

If x and y are non zero real numbers satisfying $x y (x^{2} - y^{2}) = x^{2} + y^{2}$ , find the minimum value of $x^{2} + y^{2}$ .