First slide

Cartesian plane

Question

The angle through which the coordinates axes rotated so that xy-term in the equation $5 x^{2} + 4 \sqrt{3} x y + 9 y^{2} = 0$ may be missing, is

Moderate

A

$π / 6$
B

$π / 4$
C

$π / 3$
D

$2 π / 3$

Solution

ln order to remove xy-term from the equation
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ the axes are rotate of
through an angle $θ$ given by
$\tan 2 θ = \frac{2 h}{a - b}$
Here, $a = 5, b = 9 and h = 2 \sqrt{3}$
$\begin{array}{l} ∴ \tan 2 θ = \frac{2 h}{a - b} \\ \Rightarrow \tan 2 θ = \frac{4 \sqrt{3}}{5 - 9} = - \sqrt{3} \Rightarrow 2 θ = \frac{2 π}{3} \Rightarrow θ = \frac{π}{3} \end{array}$

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