Given $f(x, y) = x^2 - 2xy + y^2$ The complete contour of the equation $f(x, y) = 1$ is described by the option(s) ___.
To solve the problem of finding the correct contour for the function \(f(x, y) = x^2 - 2xy + y^2\), given that \(f(x, y) = 1\), we need to explore the nature of the given equation.
The expression \(x^2 - 2xy + y^2\) can be rewritten by completing the square. Consider:
\(x^2 - 2xy + y^2 = (x-y)^2\)
This means that the function \(f(x, y) = (x-y)^2\). Now, to find the contour where \(f(x, y) = 1\), we simply set:
\((x - y)^2 = 1\)
Solving this equation results in two possibilities:
- \(x - y = 1\)
- \(x - y = -1\)
These are linear equations representing lines in the coordinate plane.
Thus, the complete contour described by the equation \(f(x, y) = 1\) consists of two lines: \(x - y = 1\) and \(x - y = -1\).
Justifying the exclusion of other options:
- A circle would require an equation of the form \((x-k)^2 + (y-h)^2 = r^2\), which is not the case here.
- An ellipse would have an equation involving two squared terms added, not a perfect square like we have.
Hence, the correct options are:
- A line, \(x - y = 1\)
- A line, \(x - y = -1\)
Important Questions from Algebra
- For positive non-zero real variables $x$ and $y$, if
$ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)]$
then, the value of $\frac{x}{y} + \frac{y}{x}$ is - It is given that $x$ and $y$ are integers in the following equation:
$$(x + y - 7)^2 + (y + 3x - 13)^2 = 0$$
The value of $(x^3 + y^3)$ is ________ (in integer). - If $pqr \neq 0$ and $p^{-x} = \frac{1}{q}$, $q^{-y} = \frac{1}{r}$, $r^{-z} = \frac{1}{p}$, what is the value of the product $xyz$?
- Two points $(4, p)$ and $(0, q)$ lie on a straight line having a slope of $3/4$. The value of $(p – q)$ is
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
