Bisectors of Angle between Two Lines

Q.

Find the distance between the parallel lines $15x+8y-34=0$ and $15x+8y+31=0$

Q. Find the equation of the right bisector of the line segment joining the points (3, 4) and ( – 1, 2).

Q.

The vertices $P,Q$ and $R$ of a triangle are $(2,1),(5,2)$ and $(3,4)$, respectively. Then, the circumcenter is

1. $\left(\frac{13}{4},-\frac{9}{4}\right)$

2. $\left(-\frac{13}{4},\frac{9}{4}\right)$

3. $\left(-\frac{13}{4},-\frac{9}{4}\right)$

4. $\left(\frac{13}{4},\frac{9}{4}\right)$

Q.

Distance between the lines represented by the equation $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$ is

1. 5/2

2. 5/4

3. 5

4. 0

Q. Equation of angle bisector of the lines 3x - 4y + 1 = 0 and 12x + 5y - 3 = 0 containing the point (1, 2) is

1. 99x + 27y - 2 = 0
2. 3x + 11y - 4 = 0
3. 99x - 27y – 2 = 0
4. 3x + 11y + 4 = 0

Q.

The lines represented by the equation $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$, are

1. None of these

2. Parallel

3. Perpendicular to each other

4. 

Q. Lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at R.

STATEMENT-1 : The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}.$
because

STATEMENT-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

1. Statement-1 is True, Statement -2 is False
   
2. Statement-1 is False, Statement-2 is True.
   
3. Statement- 1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
4. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Q.

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

1. 

2. 

3. 

4. None of these

Q.

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

1. 

2. 

3. 

4. None of these

Q.

The equation of the pair of straight lines through origin, each of which makes as angle $\alpha$ with the line y = x, is

1. 

2. 

3. 

4. 

Q.

The lines represented by the equation $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$, are

1. Perpendicular to each other

2. None of these

3. Parallel

4. 

Q. Let$P=(-1,0),Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points, Then the equation of the bisector of the angle PQR is -

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

Q. Show that the points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ are collinear.

Q.

The equation x- y = 4 and $x^2+4xy+y^2=0$ represent the sides of

1. an equilateral triangle

2. a right angled triangle

3. none of these

4. an isosceles triangle

Q. The equations of the bisectors of the lines 3x-4y+7=0 and 12x-5y-8=0 are:

1. 21x+27y-131=0
2. 21x-27y+131=0
3. 99x-77y+51=0
4. 99x+77y-51=0

Q.

Find equation of the line which is equidistant from parallel lines $9x+6y-7=0$ and $3x+2y+6=0$

Q.

The equation of the bisectors of the angles between the lines represented by $x^2+2xycot\theta +y^2=0$, is

1. 

2. 

3. 

4. 

Q. By using the concept of equation of a line, prove that the three points (3, 0), ( -2, -2) and (8, 2) are collinear.

Q. Let $P=(-1,0),Q=(0,0)\text{and}R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$ is

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

Q.

The angle between the lines represented by the equation $a{x}^2+xy+b{y}^2=0$ will be ${45}^{\circ }$, if

1. None of these

2. 

3. 

4. 

Q. Solve each of the following system of equations in R.
$\frac{4}{x+1}\le 3\le \frac{6}{x+1},x>0$

Q.

Equation of a straight line passing through the point $(4,5)$ and equally inclined to the lines $3x=4y+7$ and $5y=12x+6$ is

1. $9x-7y=1$

2. $9x+7y=71$

3. $7x+9y=73$

4. $7x-9y+17=0$

Q. Find the distance between the following pairs of points:
(i) $(2,3,5)$ and $(4,3,1)$
(ii)$(-3,7,2)$and $\left(\right(2,4,-1)$
(iii) $(-1,3,-4)$ and $(1,-3,4)$
(iv) $(2,-1,3)$ and $(-2,1,3)$

Q. Find the locus of the curve represented by $x=sec\theta +1andy=tan\theta -1$, where $\theta$ is a variable.

1. (x-1)² + (y+1)² = 1

2. (x-1)² - (y+1)² = 1

3. (x+1)² - (y-1)² = 0

4. (x+1)² + (y-1)² = 1

Q. Solve each of the following system of equations in R.

23. $\left|\frac{3x-4}{2}\right|\le \frac{5}{12}$

Q. Solve each of the following system of equations in R.

27. $\left|\frac{2x-1}{x-1}\right|>2$