Question

The equation $(2cos\theta +3sin\theta )x+(3cos\theta -5sin\theta )y-(5cos\theta -2sin\theta )=0$ pass through a fixed point for all values of $\theta$.

• A. True
• B. False

Answer 1

The correct option is A True

As the line passes through a fixed point, the point should satisfy the equation of line as well.

$(2\cos \theta +3\sin \theta )x+(3\cos \theta -5\sin \theta )y-(5\cos \theta -2\sin \theta )=0$

Let $x=0,y=0$

$\Rightarrow (2\cos \theta +3\sin \theta )\times 0+(3\cos \theta -5\sin \theta )\times 0-(5\cos \theta -2\sin \theta )\ne 0$

Hence L.H.S$\ne$R.H.S

Let $x=1,y=1$

$\Rightarrow (2\cos \theta +3\sin \theta )\times 1+(3\cos \theta -5\sin \theta )\times 1-(5\cos \theta -2\sin \theta )=0$

$\Rightarrow 2\cos \theta +3\sin \theta +3\cos \theta -5\sin \theta -5\cos \theta +2\sin \theta =0$

Hence $(1,1)$ is the fixed point through which the line passes.