A) \[{{45}^{o}}\]
B) \[{{60}^{o}}\]
C) \[{{30}^{o}}\]
D) \[{{75}^{o}}\]
Correct Answer: C
Solution :
Differentiation of \[x=2\sec \varphi \] Þ\[\frac{dx}{d\varphi }=2\sec \varphi \tan \varphi \] Differentiate, \[y=3\tan \varphi \]w.r.t. f, we get\[\frac{dy}{d\varphi }=3{{\sec }^{2}}\varphi \] \[\therefore \]Gradient of tangent \[\frac{dy}{dx}=\frac{dy/d\varphi }{dx/d\varphi }=\frac{3{{\sec }^{2}}\varphi }{2\sec \varphi \tan \varphi }\] \[\frac{dy}{dx}=\frac{3}{2}\,\text{cosec}\varphi \] .....(i) But, tangent is parallel to \[3x-y+4=0\] \[\therefore \]Gradient \[m=3\] .....(ii) By (i) and (ii), \[\frac{3}{2}\text{cosec}\varphi =3\]Þ\[\text{cosec}\varphi =2\], \[\therefore \varphi =30{}^\circ \].
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