Rectangular Cartesian Coordinate System

Q. Let the distance of a point on the line $x=3$ to the point $(1,-2)$ is twice that of from the point $(4,0)$ . Then the integral value for the ordinate is

Q.

Find the length of the perpendicular from the point $(x^1,y^1)$ to the straight line Ax + By + C = 0, the axes being inclined at an angle $\omega$, and the equation being written such that C is a negative quantity.

1. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.cos\omega$

2. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.sin\omega$

3. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABsin\omega }}.cos\omega$

4. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABcos\omega }}.sin\omega$

Q. Let $A(\alpha ,\beta ),B({\alpha }^2,\beta ),C(\alpha ,{\beta }^2)$ are three ditinct points which are at same distance from origin. Then the sum of all possible value of ${}^{\prime }{\theta }^{\prime }$ such that $(\sin \theta ,\cos \theta )$ is equidistant to any of these points taken pairwise is
where $(0\le \theta \le \frac{\pi }{2})$

1. $\frac{\pi }{4}$
2. $\frac{3\pi }{4}$
3. $\frac{\pi }{2}$
4. $0$