Question

Let P = $\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$ Consider the set S of all vectors $\left[\begin{array}{c}x\\ y\end{array}\right]$ such that $a^2+b^2$ = 1 where $\left(\begin{array}{c}a\\ b\end{array}\right)$ = P$\left(\begin{array}{c}x\\ y\end{array}\right)$ then s is

• A. A circle of radius $\sqrt{10}$
• B. A circle of radius $\frac{1}{\sqrt{10}}$
• C. An ellipse with major axis along $\left[\begin{array}{c}1\\ 1\end{array}\right]$
• D. An ellipse with minor axis along $\left[\begin{array}{c}1\\ 1\end{array}\right]$

Answer 1

The correct option is D An ellipse with minor axis along $\left[\begin{array}{c}1\\ 1\end{array}\right]$

P = $\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$

then,
$\Rightarrow$ $\left[\begin{array}{c}a\\ b\end{array}\right]$ = P $\left[\begin{array}{c}x\\ y\end{array}\right]$


$\Rightarrow$ $\left[\begin{array}{c}a\\ b\end{array}\right]$ = $\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$

$\Rightarrow$ 3x + y = a and x + 3y = b

Now, $a^2+b^2$ = 1

$\Rightarrow$ $(3x+y{)}^2+(x+3y{)}^2$ = 1

or $10x^2+10y^2+12xy$ = 1 .....(i)

Now comparing it with $a{x}^2+b{y}^2+2hxy=1,$ we get

a = 10, b = 10, h = 6

$\because$ $h^2$ -ab = $(6{)}^2$ - (10)(1) = -64 < 0

So (i) represents an ellipse with major axis is along y = -x = $\left[\begin{array}{c}1\\ -1\end{array}\right]$ and minor axis is along y = x = $\left[\begin{array}{c}1\\ 1\end{array}\right]$ So, option (d) is correct.