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
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
The relationship between two variables $x$ and $y$ is given by $x + py + q = 0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
The real variables $x, y, z$ and the real constants $p, q, r $ satisfy
$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$
Given the denominators are non-zero, the value of $px + qy + rz$ isThe complex function
$e^{-\left(\frac{2}{z-1}\right)}$
has __________________Consider two matrices: $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$.
Which of the following statement is/are true?
