Bisectors of Angle between Two Lines

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. Area of the triangle formed by the line $x+y=3$ and angle bisectors of the pair of straight lines $x^2-y^2+2y=1$ is

1. 2 sq. units
2. 4 sq. units
3. 6sq. units
4. 8 sq. units

Q.

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

1. $x^2-y^2$ = 0

2. $x^2-y^2$ = xy

3. $(x^2-y^2)cot\theta$ = 2xy

4. $x^2-y^2$ = 3xy

Q. The equation of the bisector of the angle between the lines $2x+y-6=0$ and $2x-4y+7=0$ which contains the point $(1,2)$, is

1. $6x-2y-5=0$
2. $2x+6y-19=0$
3. $6x+2y-5=0$
4. $2x+6y+19=0$

Q. $A(1,3)$ and $C(-2/5,-2/5)$ are the vertices of a triangle $ABC$ and the equation of the internal angle bisector of $\angle ABC$ is $x+y=2$, then

1. $BC:7x-3y+4=0$
2. $BC:7x+3y+4=0$
3. $B\equiv (5/2,9/2)$
4. $B\equiv (-5/2,9/2)$

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

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

Q.

If $L_1$ and $L_2$ be the angle bisectors of two lines, then the angle between $L_1$ and $L_2$ is${90}^{\circ }$ .

1. True

2. False

Q.

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

1. $x^2-y^2$ = 0

2. $x^2-y^2$ = xy

3. $(x^2-y^2)cot\theta$ = 2xy

4. $x^2-y^2$ = 3xy

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

Q.

Find the value of $k$ for which the following system of linear equations has infinite solutions

$kx+3y=2k+1:2(k+1)x+9y=7k+1$

Q. The vertices of a triangle are $A(-1,-7),B(5,1),$and $C(1,4).$ The equation of the bisector of $\angle ABC$ is ___

1. x-7y+2=0
2. 7x+y-2=0
3. 2y-7x-5=0
4. 2x+7y+5=0

Q. The equation(s) of the angle bisectors of the lines $3x-4y+7=0$ and $12x-5y-8=0$ is/are

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

Q.

If $L_1$ and $L_2$ be the angle bisectors of two lines, then the angle between $L_1$ and $L_2$ is${90}^{\circ }$ .

1. True

2. False

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. The image of a point $(3t+1,1-4t),\forall t\in R-\left\{0\right\}$ in a line, lies on $3x-4y+1=0.$ Then the slope of line(s) is (are)

1. $7$
2. $-7$
3. $\frac{1}{7}$
4. $-\frac{1}{7}$

Q. If $A(1,3)$ and $C(-\frac{2}{5},-\frac{2}{5})$ are the vertices of a $△ABC$ and the equation of the internal angle bisector of $\angle ABC$ is $x+y=2$, then

1. the equation of line $BC$ is $7x-3y+4=0$
2. the equation of line $BC$ is $7x+3y+4=0$
3. $B\equiv (\frac{5}{2},\frac{9}{2})$
4. $B\equiv (-\frac{5}{2},\frac{9}{2})$

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. The sides of a rhombus $ABCD$ are parallel to the lines, $x-y+2=0$ and $7x-y+3=0$. If the diagonals of the rhombus intersect at $P(1,2)$ and the vertex $A$ (different from the origin) is on the $y-$axis, then the ordinate of $A$ is

1. $\frac{7}{4}$
2. $2$
3. $\frac{5}{2}$
4. $\frac{7}{2}$

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. The equation of the bisector of obtuse angle formed by the lines 3x+4y-11=0 and 12x-5y-2=0 will be .

1. 11x-3y-17=0
2. 11x+3y-17=0
3. 3x+11y-19=0
4. 3x-11y+19=0

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. an isosceles triangle

4. none of these

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

1. $x^2-y^2$ = 0

2. $x^2-y^2$ = xy

3. $(x^2-y^2)cot\theta$ = 2xy

4. $x^2-y^2$ = 3xy

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 vertices of a triangle are $A(-1,-7),B(5,1),$and $C(1,4).$ The equation of the bisector of $\angle ABC$ is ___

1. x-7y+2=0
2. 7x+y-2=0
3. 2y-7x-5=0
4. 2x+7y+5=0

Q. The equation of the bisector of obtuse angle formed by the lines 3x+4y-11=0 and 12x-5y-2=0 will be .

1. 11x-3y-17=0
2. 11x+3y-17=0
3. 3x+11y-19=0
4. 3x-11y+19=0

Q. The equation(s) of the angle bisectors of the lines $3x-4y+7=0$ and $12x-5y-8=0$ is/are

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

Q. If $A(1,3)$ and $C(-\frac{2}{5},-\frac{2}{5})$ are the vertices of a $△ABC$ and the equation of the internal angle bisector of $\angle ABC$ is $x+y=2$, then

1. the equation of line $BC$ is $7x-3y+4=0$
2. the equation of line $BC$ is $7x+3y+4=0$
3. $B\equiv (\frac{5}{2},\frac{9}{2})$
4. $B\equiv (-\frac{5}{2},\frac{9}{2})$

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 equation of the bisector of the acute angle between $4x+3y-6=0$ and $5x+12y+9=0$ is

1. $9x-7y-41=0$
2. $7x+9y-3=0$
3. $9x+7y-3=0$
4. $9x-7y+41=0$