Question

solutions?
If x, y are real numbers and (x - 5)2 + (x- y)2 = 0, then what are the values of x and
y?
​

Answer 1

We are given that two real numbers x and y satisfy the equation

y

x

=x−y.

We will discuss each option separately.

Option A)

Multiplying the above equation throughout by y and rearranging we get,

y

2

−xy+x=0. We will treat this as a quadratic equation in the variable y. The discriminant of this equation is x

2

−4x. We know that the above quadratic equation has a real solution in y if and only if the discriminant is greater than or equal to 0. That is, if and only if x

2

−4x≥0. That is, if and only if x(x−4)≥0. So we observe that the discriminant of the equation is greater than or equal to 0 if and only if x≥4 or x≤0. So option A is correct.

Option B)

If we put y=1 in the equation

y

x

=x−y, then we get x=x−1, which has no solution. So option B is wrong.

Option C) and Option D)

We claim that y=2 and x=4 is a solution for th egiven equation. For this, we write-

LHS=

y

x

=

2

4

=2, and

RHS=x−y

=4−2=2.

As 2 and 4 are integers, Option C and Option D are wrong.

Option E)

We write the equation in the form y

2

−xy+x=0.

After putting x=5, we get y

2

−5y+5=0.

By the quadratic formula, the roots of this equation are :

y=

2

5±

5

2

−20

=

2

5±

5

.

This means that x=5 and y=

2

5+

5

is a solution of the given equation in which y is