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A vector a has components $2 p$ and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If a has components $p+\mathrm{l}$ and $\mathrm{l}$ with respect to the new system, then
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Verified Answer
The correct answer is:
$p=1$ or $p=\frac{-1}{3}$
Let point $A(2 p, 1)$
A is rotated about origin in the anti-clock wise direction then new coordinate is $(p+1,1)$
Squaring and adding Eqs. (i) and (ii), we get
$\begin{gathered}
4 p^2+1=(p+1)^2+1 \Rightarrow 4 p^2=p^2+2 p+1 \\
\Rightarrow 3 p^2-2 p-1=0 \Rightarrow(p-1)(3 p+1)=0 \Rightarrow p=1, \frac{-1}{3}
\end{gathered}$
A is rotated about origin in the anti-clock wise direction then new coordinate is $(p+1,1)$
Squaring and adding Eqs. (i) and (ii), we get
$\begin{gathered}
4 p^2+1=(p+1)^2+1 \Rightarrow 4 p^2=p^2+2 p+1 \\
\Rightarrow 3 p^2-2 p-1=0 \Rightarrow(p-1)(3 p+1)=0 \Rightarrow p=1, \frac{-1}{3}
\end{gathered}$
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