Perpendicular Form of a Straight Line

Q. A line passes through $(2,3)$ and the portion of it intercepted between co-ordinate axes is divided in the ratio $2:3$ at the point $(2,3).$ Then equation of the line in Normal form is

1. $x\cos \frac{\pi }{4}+y\sin \frac{\pi }{4}=\frac{5}{\sqrt{2}}$
2. $x\cos \frac{\pi }{3}+y\sin \frac{\pi }{3}=\frac{3}{\sqrt{2}}$
3. $\frac{9}{\sqrt{97}}x+\frac{4}{\sqrt{97}}y=\frac{30}{\sqrt{97}}$
4. $\frac{3}{\sqrt{13}}x+\frac{2}{\sqrt{13}}y=\frac{6}{\sqrt{13}}$

Q. The equation of the line perpendicular to the line $2x+3y+5=0$ and passing through $(1,1)$, is

1. $3x-2y+1=0$
2. $3x-2y=2$
3. $3x-2y=1$
4. $3x-2y+2=0$

Q. Let $PQRS$ is a parallelogram where $P=(2,2),Q=(6,-1),\text{and}R=(7,3)$. Then equation of the line through $S$ and perpendicular to $QR$ is

1. $x+4y=27$
2. $4x-3y+6=0$
3. $x+4y=10$
4. $x+4y=2$

Q. Consider two points $A(1,a)$ and $B(5,b)$. If the equation of the line bisecting the line segment $AB$ perpendicularly is $x-3y=0$ then $|ab|$ is

Q. The perpendcular form of the straight line
$y=x+4$ is $x\cos \alpha +y\sin \alpha =p$ then

1. $\alpha ={45}^{\circ }$
2. $\alpha ={135}^{\circ }$
3. $p=2\sqrt{2}$
4. $p=\sqrt{2}$

Q. If two lines perpendicular to each other always satisfy the condition $\cot {\theta }_1+\cot {\theta }_2=2,$ where ${\theta }_1$ and ${\theta }_2$ are the angles made by the lines with positive direction of $X-$axis and $M_1,$ $M_2$ be the respective slopes, then $|M_1+\sqrt{2}{M}_2|$ is equal to
(where $M_1>M_2$)

Q. The perpendicular distance of a line from origin is 2 units and the perpendicular makes an angle α with X-axis such that sinα=13. The equation of line is .

1. $2\sqrt{2}x+y=6$
2. $2\sqrt{2}x-y=6$
3. $x+2\sqrt{2}y=6$
4. $x-2\sqrt{2}y=6$

Q. A square of side length $a$ units lies above the $x-$axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha (0<\alpha <\pi /4)$ with the positive direction of $x-$axis. The equation of its diagonal not passing through the origin is

1. $y(\cos \alpha +\sin \alpha )+x(\sin \alpha -\cos \alpha )=a$
2. $y(\cos \alpha +\sin \alpha )+x(\sin \alpha +\cos \alpha )=a$
3. $y(\cos \alpha +\sin \alpha )+x(\cos \alpha -\sin \alpha )=a$
4. $y(\cos \alpha -\sin \alpha )-x(\sin \alpha -\cos \alpha )=a$

Q. The line perpendicular to $y=\sqrt{3}x+2$ and passing through (2, 1) is rotated through an angle ${60}^{\circ }$ about (2, 1). Area of triangle formed by these lines and $x$-axis is $A{\text{units}}^2$, then $4A^2$ is :