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  • The acute angle between the line 3 x-4 y=5 and the circle <img alt="\mathrm{x^2+y^2-4 x+2 y-4=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2-4%20x&am

The acute angle between the line 3 x-4 y=5 and the circle \mathrm{x^2+y^2-4 x+2 y-4=0} is \mathrm{\theta}, then find the value of \mathrm{\sin \theta}.

Option: 1

\frac{2 \sqrt{2}}{3}


Option: 2

\frac{ \sqrt{2}}{3}


Option: 3

\frac{ \sqrt{3}}{2}


Option: 4

\frac{3}{2 \sqrt{2}}


Answers (1)

best_answer

We have, the circle with centre (2, –1) and radius = 3 Perpendicular distance from centre on 3x – 4y = 5 is given by

\mathrm{\begin{aligned} & p=\left|\frac{6+4-5}{5}\right|=1 \\ & \therefore \quad \sin \left(90^{\circ}-\theta\right)=\frac{1}{3} \\ & \Rightarrow \cos \theta=1 / 3 \\ & \Rightarrow \sin \theta=\frac{2 \sqrt{2}}{3} \end{aligned}}

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