Bisectors of Angle between Two Lines

Q. Consider the following statements relating to 3 lines $L_1$, $L_2$ and $L_3$ in the same plane
(1). If $L_2$ and $L_3$ are both parallel to $L_1$, then they are parallel to each other.
(2). If $L_2$ and $L_3$ are both perpendicular to $L_1$, then they are parallel to each other.
(3). If the acute angle between $L_1$ and $L_2$ is equal to to acute angle between $L_1$ and $L_3$, then $L_2$ is parallel to $L_3$.

1. (1) and (2) are correct
2. (2) and (3) are correct
3. (1) and (3) are correct
4. (1), (2) and (3) are correct

Q.

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

1. a right angled triangle

2. an isosceles triangle

3. none of these

4. an equilateral triangle

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

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

Q. The equation of the plane bisecting the angle between the planes $3x+4y=4$ and $6x-2y+3z+5=0$ that contains the origin, is

1. $9x-38y+15z+43=0$
2. $51x+18y+15z=3$
3. $17x+9y+15z=26$
4. $9x+2y+3z+1=0$

Q. Assertion :The ratio $PR:RQ$ equal $2\sqrt{2}:\sqrt{5}$ Reason: In nay triangle, bisector of an angle divides the triangle into two similar triangles.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. The co-ordinates of two points $P$ and $Q$ are $(x_1,y_1)$ and $(x_2,y_2)$ and $O$ is the origin. If circles be described on $OP$ and $OQ$ as diameters then length of their common chord is

1. $\frac{|x_1{x}_2+y_1{y}_2|}{PQ}$
2. $\frac{|x_1{x}_2+y_1{y}_2|}{PQ}$
3. $\frac{|x_1{y}_2+x_2{y}_1|}{PQ}$
4. $\frac{|x_1{y}_2-x_2{y}_1|}{PQ}$