Question

Let x and y be real numbers such that 4x^2−5xy+4y^2=5 . The sum of the reciprocals of the maximum and the minimum values of x^2+y^2 is 8/k then k is .......

Answer 1

Step-by-step explanation:

4x^2−5xy+4y^2=5

x^2+y^2

k= 5/2

Answer 2

The value of k is 5.

Given:

Let,  x and y be real numbers such that;

$4x^{2} -5xy+4y^{2}=5$

To Find:

The sum of the reciprocals of the maximum and the minimum values of    ( $x^{2} +y^{2}$ )is 8/k then the value of k is =?

Solution:

We have the given equation as follows:

$4x^{2} -5xy+4y^{2}=5$

Now, By simplifying the above equation, we get;

⇒  $4x^{2} +4y^{2} = 5+5xy$

⇒  $4(x^{2}+y^{2} ) = 5(1+xy)$

⇒  $x^{2} +y^{2} = \frac{5}{4}(1+xy)$                         --------------(1)

Here, we need the maximum and minimum values of the equation            ( $x^{2} +y^{2}$)  to solve the question, i.e., we need to find the maximum values of  $\frac{5}{4}(1+xy)$ .

We know that, If the Arithmetic mean of the two numbers is AM and the geometric mean of the two numbers is GM, then;

AM ≥ GM always holds.

⇒  $\frac{x+y}{2}$  ≥  $\sqrt{xy}$

⇒  $(x+y)$  ≥  $2\sqrt{xy}$

By squaring both sides, we get;

⇒  $x^{2}+y^{2}+2xy$  ≥  $4xy$

⇒   $x^{2} +y^{2}$  ≥  $2xy$

From equation (1), we can write;

$\frac{5}{4} (1+xy)$  ≥  $2xy$

∴  $xy$  ≤  $\frac{5}{3}$

i.e., the Maximum value of xy is  $\frac{5}{3}$ .

Thus, we can get the maximum value  $x^{2} +y^{2}$  as follows;

$(x^{2} +y^{2} )_{Max}$  = $\frac{5}{4}(1+\frac{5}{3} )$

∴  $(x^{2} +y^{2} )_{Max}$  = $\frac{10}{3}$

Similarly, By modifying the equation (1) as follows, we get;

$x^{2} +y^{2}$  =  $(x+y)^{2}$ - 2xy = $\frac{5}{4}(1+xy)$

⇒  $(x+y)^{2}$ = $\frac{5}{4}+\frac{13}{4}xy$  ≥  0

⇒  xy ≥ $-\frac{5}{13}$

∴ $(x^{2} +y^{2} )_{Min}$  =  $\frac{5}{4} (1-\frac{5}{13} )$

∴ $(x^{2} +y^{2} )_{Min}$  =  $\frac{10}{13}$

Now, by adding the reciprocals of the maximum  and minimum values of ( $x^{2} +y^{2}$ ) we get 8/5 and comparing the answer with 8/k;

we get the value of k as 5.

Therefore, The sum of the reciprocals of the maximum and the minimum values of    ( $x^{2} +y^{2}$ )is 8/k then the value of k is 5.

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