Condition for Two Lines to Be Parallel

Q. Find the values of k for which the line is (a) Parallel to the x -axis, (b) Parallel to the y -axis, (c) Passing through the origin.

Q.

The equation $8x^2+8xy+2y^2+26x+13y+15=0$ represents a pair of straight lines.The distance between them is

1. 

2. 

3. 

4. 

Q.

Solve the following systems of inequalities graphically:

$2x+y\ge 4,x+y\le 3,2x-3y\le 6$

Q.

Find the equation of the bisectors of the angle between the lines represented by $3x^2-5xy+4y^2=0$

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

2. $5x^2-2xy+5y^2=0$

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

4. $5x^2-2xy-5y^2=0$

Q.

A rectangle ABCD, A = (0, 0), B = (4, 0), C = (4, 2), D = (0, 2) undergoes the following transformations successively.
(i) $f_1(x,y)\to (y,x)$
(ii) $f_2(x,y)\to (x+3y,y)$
(iii) $f_3(x,y)\to (\frac{x-y}{2},\frac{x+y}{2})$
The final figure will be

1. a square

2. a rectangle

3. a parallelogram

4. a rhombus

Q. The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is

(a) 90°
(b) 60°
(c) 45°
(d) 30°
(e) 180°

Q.

Find the distance between parallel lines

(i) 15x + 8y - 34 = 0 and 15x + 8y + 31 = 0

(ii) l(x + y) + p = 0 and l(x + y) - r = 0

Q. Find equation of the line parallel to the line 3 x – 4 y + 2 = 0 and passing through the point (–2, 3).

Q. Write the following intervals in set-builder form:
(i) $(-3,0)$

Q. Find equation of the line through the point (0, 2) making an angle with the positive x -axis. Also, find the equation of line parallel to it and crossing the y -axis at a distance of 2 units below the origin.

Q. 8. In a plane there are 27 straight lines, of which 13 pass through the point A and 11 pass through point B. Bedsides, no three lines pass through points A and B and no two are parallel. Find the number of points of intersection of the straight lines.

Q. If the line 2x+3y=5 and y=mx+c be parallel, then

1. $m=-\frac{2}{3},c$ = any real number
   
2. $m=\frac{2}{3},c=5$
3. $m=-\frac{2}{3},c=5$
4. None of these

Q. Let PS be the median of the triangle with vertices P(2, 2), Q(6, -1) and R(7, 3). The equation of the line passing through (1, -1) and parallel to PS is

1. 2x – 9y – 7 = 0
2. 2x – 9y + 7 = 0
3. 2x – 9y – 11 = 0
4. 2x + 9y – 7 = 0

Q. If $AB\parallel CD,EF\parallel GH$, then which of the following is true for measure of $\angle a,\angle b,\angle c$ ?

1. ${114}^{\circ },{46}^{\circ },{114}^{\circ }$
2. ${44}^{\circ },{116}^{\circ },{114}^{\circ }$
3. ${112}^{\circ },{46}^{\circ },{44}^{\circ }$
4. None of these

Q. Let $A=\{1,2\}$ and $B=\{3,4\}$. Write $A\times B$ and find how many subsets will $A\times B$ have? List them.

Q. Without using distance formula, show that points $(-2,-1),(4,0),(3,3)$ and $(-3,2)$ are the vertices of a parallelogram

Q.

The combined equation of the bisectors of the angle between the lines represented by $(x^2+y^2)\sqrt{3}=4xy$ is

1. 

2. 

3. 

4. 

Q. Prove that the lines 2x − 3y + 1 = 0, x + y = 3, 2x − 3y = 2 and x + y = 4 form a parallelogram.

Q.

Let PS, be the median of the triangle with vertices P(2, 2), Q(6, -1) and R(7, 3) . Find the equation of line passing through (-1, 1) and parallel to PS.

Q.

If PS is the median of the triangle with vertices P(2, 2), Q(6, -1) and R(7, 3) then equation of the line passing through (1, -1) and parallel to PS is

1. $4x-7y-11=0$

2. $2x+9y+7=0$

3. $4x+7y+3=0$

4. $2x-9y-11=0$

Q. In the given figure, if $r\parallel s,p\parallel q$ and $u\parallel t$, then $c$ equals

1. $a-b$
2. $2a+b$
3. $a+b$
4. $b-a$

Q. Consider the set of all lines $px+qy+r=0$ such that $3p+2q+4r=0$. Which of the following statement is true?

1. The lines are all parallel.
2. The lines are concurrent at the point $(\frac{3}{4},\frac{1}{2})$.
3. The lines are not concurrent.
4. Each line passes through the origin.

Q. If $2ax-2y+3z=0,x+ay+2z=0\text{and}2x+az=0$ has a non-trivial solution, then the value of $a$ is:

1. $1$
2. $2$
3. $4$
4. $0$

Q. If two lines having slopes $m_1$ and $m_2$ are parallel to each other, then which of the following is correct?

1. $m_1=m_2$
2. $m_1\times m_2=1$
3. $\frac{m_1}{m_2}=2$
4. $m_1=3\times m_2$

Q.

A rectangle ABCD, A = (0, 0), B = (4, 0), C = (4, 2), D = (0, 2) undergoes the following transformations successively.
(i) $f_1(x,y)\to (y,x)$
(ii) $f_2(x,y)\to (x+3y,y)$
(iii) $f_3(x,y)\to (\frac{x-y}{2},\frac{x+y}{2})$
The final figure will be

1. a rhombus

2. a rectangle

3. a square

4. a parallelogram

Q. Prove that the line through the point ( x 1 , y 1 ) and parallel to the line A x + B y + C = 0 is A ( x –x 1 ) + B ( y – y 1 ) = 0.

Q.

Equation of a line which is parallel to the line common to the pair of lines given by $6x^2-xy-12y^2-0$ and $15x^2+14xy-8y^2-0$ and at a distance 7 from it is

1. $3x+4y=35$

2. $5x-2y=7$

3. $3x+4y=-35$

4. $2x-3y=7$

Q.

Prove that the line through the point $(x_1,y_1)$ and parallel to the line Ax + By + C = 0 is $A(x-x_1)+B(y-y_1)=0$.

Q. Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan⁻¹$\left(\frac{a}{b}\right)$.

Q. Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.