Question 2016.S2.32: Let P = [ 3 1 1 3 ] . Consider the set S of all vectors (x y......
Let P=\left[\begin{array}{ll}3 & 1 \\1 & 3\end{array}\right] . Consider the set S of all vectors \left\lgroup \begin{array}{l}x \\y\end{array} \right\rgroup such that a²+ b²= 1 where \left\lgroup \begin{array}{l}a \\b\end{array} \right\rgroup = P\left\lgroup \begin{array}{l}x \\y\end{array} \right\rgroup . 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\lgroup \begin{array}{l}1 \\1\end{array} \right\rgroup
(d) an ellipse with minor axis along \left\lgroup \begin{array}{l}1 \\1\end{array} \right\rgroup
Given that
\left[\begin{array}{ll}3 & 1 \\1 & 3\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}a \\b\end{array}\right]
Then
\begin{aligned}& 3 x+y=a \quad x+3 y=b \\& a^2+b^2=1\end{aligned}
Therefore,
\begin{aligned}& (3 x+y)^2+(x+3 y)^2=1 \\& 10 x^2+10 y^2+12 x y=1 \\& \Rightarrow a=10, b=10 \text { and } h=0\end{aligned}
This represents an ellipse. Then length of semi-axis is
\begin{aligned}& \left(a b-h^2\right) r^4-(a+b) r^2+1=0 \\& r=\frac{1}{2} \text { or } \frac{1}{4}\end{aligned}
Length of minor axis =2 r=2\left\lgroup \frac{1}{4} \right\rgroup=\frac{1}{2}
Equation of minor axis is
\begin{aligned}& \left\lgroup a-\frac{1}{r^2} \right\rgroup x+h y=0 \\& (10-16) x+6 y=0 \\& y-x=0\end{aligned}
Similarly major axis equation is y + x = 0
So, \left[\begin{array}{l}1 \\1\end{array}\right] lies on the minor axis.