The set of all possible values of theta in th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the line be $L(x, y) = x+y-1 = 0$. For the points $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (\sin 	heta, \cos 	heta)$ to lie on the same side

Vedclass · Accepted Answer

$\left(0, \frac{\pi}{2}\right)$

Vedclass · Answer

(A) Let the line be $L(x, y) = x+y-1 = 0$. For the points $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (\sin 	heta, \cos 	heta)$ to lie on the same side of the line,the product $L(x_1, y_1) \cdot L(x_2, y_2)$ must be greater than $0$. $L(1, 2) = 1+2-1 = 2 > 0$. Therefore,we must have $L(\sin 	heta, \cos 	heta) > 0$. $\Rightarrow \sin 	heta + \cos 	heta - 1 > 0$ $\Rightarrow \sin 	heta + \cos 	heta > 1$ Dividing by $\sqrt{2}$,we get: $\frac{1}{\sqrt{2}} \sin 	heta + \frac{1}{\sqrt{2}} \cos 	heta > \frac{1}{\sqrt{2}}$ $\Rightarrow \sin \left(	heta + \frac{\pi}{4}ight) > \frac{1}{\sqrt{2}}$ Since $	heta \in (0, \pi)$,we have $	heta + \frac{\pi}{4} \in \left(\frac{\pi}{4}, \frac{5\pi}{4}ight)$. In this interval,$\sin \left(	heta + \frac{\pi}{4}ight) > \frac{1}{\sqrt{2}}$ holds when: $\frac{\pi}{4} Subtracting $\frac{\pi}{4}$ from all sides,we get: $0 Thus,the set of values is $\left(0, \frac{\pi}{2}ight)$.

The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1, 2)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $x+y=1$,is . . . . . . .

Similar Questions

If $p$ is the length of the perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$,then $\frac{1}{a^2}+\frac{1}{b^2}=$

Find the distance of the point $(2, 3)$ from the line $2x - 3y + 9 = 0$ measured along the direction of the line $x - y + 1 = 0$.

Find the distance between the lines $3x + 4y + 7 = 0$ and $3x + 4y + 22 = 0$.

If $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line $x+y+3=0$ such that each of them is at a distance of $\sqrt{5}$ units from the line $x+2y+2=0$,then the value of $|x_1-x_2|$ is:

$A$ variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $A$ and $B$. The minimum area of the triangle $OAB$,where $O$ is the origin,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module