Angle between Two Lines

Q.

Angle between the lines represented by the equation $x^2+2xysec\theta +y^2=0,$is

1. $\theta$

2. $2\theta$

3. $\frac{\theta }{2}$

4. none of these

Q. The equations of two equal sides of an isosceles triangle are 7x – y + 3 = 0 and x + y –3 = 0 and the third side passes through the point (1, -10). The equation of the third side is

1. 3x + y + 7 = 0 but not x – 3y – 31 = 0
2. 3x + y + 7 = 0 or x – 3y – 31 = 0
3. Neither 3x + y + 7 nor x – 3y – 31 = 0
4. x – 3y – 31 = 0 but not 3x + y + 7 = 0

Q.

If the angle between lines is represented by an equation ${\mathrm{y}}^2+\mathrm{kxy}-{\mathrm{x}}^2{\tan }^2\mathrm{A}=0\mathrm{is}2\mathrm{A}$ then$k=$

1. $0$

2. $1$

3. $2$

4. $\tan A$

Q.

Two lines passing through the point (2, 3) intersects each other at an angle of ${60}^{\circ }$. If slope of one line is 2; Find equation of the other line.

Q. Find angles between the lines

Q.

Match the following:
[z represents a curve or collection of points in each case]
$\begin{array}{cc}p)z\overline{z}+a\overline{z}+\overline{a}z+b=0,a\overline{a}-b>0& \left(1\right)straightline\\ \left(q\right)z\overline{a}+\overline{z}a+b=0& \left(2\right)hyperbola\\ \left(r\right)arg(z-z_0)=\theta & \left(3\right)ellipse\\ \left(s\right)|z-z_1|+|z-z_2|=a,a>|z-z_2|& \left(4\right)circle\end{array}$

1. P - 4, Q - 1, R - 1, S - 3

2. P - 3, Q - 1, R - 1, S - 2

3. P - 4, Q - 1, R - 4, S - 3

4. P - 4, Q - 1, R - 4, S - 2

Q.

The complex numbers z = x + iy which satisfy the equation $|$ $\frac{z-4i}{z+4i}$ $|$ = 1, lie on

1. Circle

2. z + = 0

3. z - = 0

4. x - axis

Q. Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point

1. $(2,1)$
2. $(2,2)$
3. $(1,2)$
4. $(1,3)$

Q. Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.

Q.

The triangle joining the points P(2, 7), Q(4, -1), R(-2, 6) is

1. Equilateral triangle

2. Right-angled triangle

3. Isosceles triangle

4. Scalene triangle

Q.

If $r(1-m^2)+m(p-q)=0$, then a bisector of the angle between the lines represented by the equation $p{x}^2-2rxy+q{y}^2=0$, is

1. 

2. 

3. 

4. 

Q.

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Q. The slope of a line is double of the slope of another line. If tangent of the angle between them is , find the slopes of he lines.

Q. The angle between the two lines whose direction cosines are $(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2})$ and $(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{-\sqrt{3}}{2})$

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

Q. Equation of the straight line passing through point of intersection of the line $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ and is making an angle $\frac{\pi }{4}$ with the line $2x-2y+3=0$ is

1. $(a+b)x=ab$
2. $(a+b)x+ab=0$
3. $(a+b)y=ab$
4. $aby=a+b$

Q. The foot of the perpendicular of (3, 6) on the line $x-2y+4=0$ is

1. (4, 4)
2. (2, 3)
3. (5, 2)

Q.

The angle between the lines represented by the equation $x^2-2pxy+y^2=0$, is

1. None of these

2. 

3. 

4. 

Q. The equation of two equal sides of an isosceles triangle are $7x-y+3=0$ and $x+y-3=0.$ If the third side passes through $(1,-10)$ and its slope is negative, then the equation of the third side is

1. $3x+y+7=0$
2. $2x+y+8=0$
3. $3x+y-7=0$
4. $x-3y-31=0$

Q.

Find the value of x-intercept made by the circle $x^2+y^2-14x+9y+45=0$

___

Q. Show that the equation of the line passing through the origin and making an angle $\theta$ with the line $y=mx+c$ is $\frac{y}{x}=\frac{m\pm \tan \theta }{1\mp m\tan \theta }$

Q. The equation of straight line equally inclined to the axes and equidistant from the point $(1,2)$ and $(3,4)$ is

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

Q.

If the bisector of the angles between the pairs of lines given by the equation
$a{x}^2+2hxy+b{y}^2=0$ and $a{x}^2+2hxy+b{y}^2+\lambda (x^2+y^2)=0$
be coincident, then $\lambda$ =

1. b

2. h

3. Any real number

4. a

Q.

A straight line L through the point (3, -2) is inclined at an angle ${60}^{\circ }$ to the line $\sqrt{3}x+y=1$. If L also intersects the X-axis, then the equation of L is

1. $y+\sqrt{3}x+2-3\sqrt{3}=0$

2. $y-\sqrt{3}x+2+3\sqrt{3}=0$

3. $\sqrt{3}y+x-3+2\sqrt{3}=0$

4. $\sqrt{3}y-x+3+2\sqrt{3}=0$

Q. If a line is equally inclined with the co-ordinate axes and also equally inclined with $2x-y+7=0$ and $ky-k^{2}x+y-2=0$, then the possible number of value(s) of $k$ is

Q.

The angle between the lines xy = 0 is

1. 

2. 

3. 

4. 

Q. The equations of two equal sides of an isosceles triangle are 7x – y + 3 = 0 and x + y –3 = 0 and the third side passes through the point (1, -10). The equation of the third side is

1. Neither 3x + y + 7 nor x – 3y – 31 = 0
2. 3x + y + 7 = 0 but not x – 3y – 31 = 0
3. x – 3y – 31 = 0 but not 3x + y + 7 = 0
4. 3x + y + 7 = 0 or x – 3y – 31 = 0

Q. A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of L is

1. $y-\sqrt{3}x+2+3\sqrt{3}=0$
2. $\sqrt{3}y-x+3+2\sqrt{3}=0$
3. $\sqrt{3}y+x-3+2\sqrt{3}=0$
4. $y+\sqrt{3}x+2-3\sqrt{3}=0$

Q.

Let $PQR$ be a right angled isosceles triangle right angled at $P(2,1)$. If the equation of the line $QR$ is $2x+y=3$, then the equation representing the pair of lines $PQ$ and $PR$ is

1. $3x^2-3y^2+8xy+20x+10y+25=0$

2. $3x^2-3y^2+8xy-20x-10y+25=0$

3. $3x^2-3y^2+8xy+10x+15y+20=0$

4. $3x^2-3y^2-8xy-10x-15y-20=0$

Q. The base of the pyramid $AOBC$ is an equilateral triangle $OBC$ with each side equal to $4\sqrt{2},$ where $O$ is the origin of reference, $AO$ is perpendicular to the plane of $△OBC$ and $|\overrightarrow{AO}|=2.$ Then the cosine of the angle between the skew straight lines, one passing through $A$ and the mid-point of $OB$ and the other passing through $O$ and the mid-point of $BC,$ is

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

Q. Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2 y = 3.