Perpendicular Form of a Straight Line

Q.

The angle between the lines $\sqrt{3}x-y-2=0$ and $x-\sqrt{3}y+1=0$ is

1. $90°$

2. $60°$

3. $45°$

4. $30°$

Q.

The acute angle between the lines $y=3$ and $\sqrt{3}x+9$ is:

1. $30°$

2. $60°$

3. $45°$

4. $90°$

Q.

The lines represent by $a{x}^2+2hxy+b{y}^2=0$ are perpendicular to each other, if

1. $h^2=a+b$

2. $a+b=0$

3. $h^2=ab$

4. $h=0$

Q.

The equation of the line, which bisects the line joining two points $(2,-19)$ and $(6,1)$ and perpendicular to the line joining two points $(-1,3)$ and $(5,-1)$ is:

1. $3x-2y=30$

2. $2x-y-3=0$

3. $2x+3y=20$

4. None of these

Q.

The length of the perpendicular from the origin to a line is 7 and the line makes an angle of ${150}^{\circ }$ with the positive direction of y-axis. Find the equation of the line.

Q. A straight line $L$ at a distance of $4$ units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of ${60}^{\circ }$ with the line $x+y=0.$ Then an equation of the line $L$ is :

1. $(\sqrt{3}+1)x+(\sqrt{3}-1)y=8\sqrt{2}$
2. $(\sqrt{3}-1)x+(\sqrt{3}+1)y=8\sqrt{2}$
3. $\sqrt{3}x+y=8$
4. $x+\sqrt{3}y=8$

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

Q.

Reduce the equation $\sqrt{3}x+y+2=0$ to:

(i) slope-intercept form and find slope and y-intercept;

(ii) intercept form and find intercept on the axes;

(iii) the normal form and find p and $\alpha .$

Q.

The angle between the straight lines $x-y\sqrt{3}=5$ and $x\sqrt{3}+y=7$ is

1. ${60}^{°}$

2. ${90}^{°}$

3. ${30}^{°}$

4. ${75}^{°}$

Q.

A line meets the x-axis and y-axis at points $AandB$ respectively. If the middle point of $ABbe(x_{1_{}},y_{1_{}})$, then the equation of the line is

1. $y_{1}x+x_{1}y=2x_1{y}_1$

2. $x_{1}x+y_{1}y=2x_1{y}_1$

3. $y_{1}x+x_{1}y=x_1{y}_1$

4. $x_{1}x+y_{1}y=x_1{y}_1$

Q.

Find the value of $\theta$ and p, if the equation $xcos\theta +ysin\theta =p$ is the normal form of the line $\sqrt{3}x+y+2=0.$

Q. The locus of feet of perpendiculars drawn from the origin to the straight lines passing through $(2,1)$ is

1. $x^2+y^2+2x+y=0$
2. $x^2+y^2-5y=0$
3. $x^2+y^2-2x-y=0$
4. $2x+y-5=0$

Q.

Find the equation of the perpendicular bisector of the line segment joining points $(7,1)$ and $(3,5)$

Q.

Find the equation of the straight line upon which the length of the perpendicular from the origin is 2 and the slope of this perpendicular is $\frac{5}{12}.$

Q.

Reduce the following equations to the normal form and find p and $\alpha$ in each case :

$\left(i\right)x+\sqrt{3}y-4=0$ $\left(ii\right)x+y+\sqrt{2}=0$ $\left(iii\right)x-y+2\sqrt{2}=0$ $\left(iv\right)x-3=0$ $\left(v\right)y-2=0$

Q.

Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x-axis is ${15}^{\circ }.$

Q.

Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is ${30}^{\circ }.$

Q.

Find the equation of the straight line on which the length of the perpendicular from the origin is 2 and the perpendicular makes an angle $\alpha$ with x-axis such that $sin\alpha =\frac{1}{3}.$

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.

The perpendicular distance of a line from the origin is 5 units and its slope is - 1. Find the equation of the line.

Q. If $L,M$ are the feet of the perpendiculars from $(2,4,5)$ to $xy-$plane and $yz-$plane respectively, then distance $LM$ (in units) is:

1. $9\sqrt{2}$
2. $2\sqrt{2}$
3. $\sqrt{29}$
4. $\sqrt{31}$

Q.

The straight line passes through the point of intersection of the straight lines $x+2y-10=0$ and $2x+y+5=0,$ is

1. $5x–4y=0$

2. $5x+4y=0$

3. $4x–5y=0$

4. $4x+5y=0$

Q.

Find the equation of a straight line on which the perpendicular from the origin makes an angle of ${30}^{\circ }$ with x-axis and which forms a triangle of area $50/\sqrt{3}$ with the axes.

Q.

Find the equation of the straight line which makes a triangle of area $96\sqrt{3}$ with the axes and perpendicular from the origin to it makes an angle of ${30}^{\circ }$ with y-axis.

Q. Let $L$ be the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$. If $L$ makes an angle $\alpha$ with the positive $X$-axis, then $\cos \alpha$ equals

1. $\frac{1}{\sqrt{3}}$
2. $\frac{1}{\sqrt{2}}$
3. $\frac{1}{\sqrt{5}}$
4. 1

Q. The equation of the line which is at a distance of $3$ units from the origin and the perpendicular from the origin makes an angle of ${30}^{\circ }$ with positive $x-$axis is

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

Q.

Find the equation of a line for which

(i) p = 5, $\alpha ={60}^{\circ }$

(ii) p = 4, $\alpha ={150}^{\circ }$

(iii) p = 8, $\alpha ={225}^{\circ }$

(iv) p = 8, $\alpha ={300}^{\circ }$

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 intercept form of the line $3x-4y+12=0$ is

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

Q.

The three lines 3x + 4y + 6 = 0, $\sqrt{2}x+\sqrt{3}y+2\sqrt{2}=0$ and 4x + 7y + 8 = 0 are

1. sides of a triangle

2. concurrent

3. parallel

4. None of these