Question 1 :
The two straight lines $a_1x+b_2y+c_2=0$ and $a_2x+b_2y+c_2z=0$ will be parallel to each other, if
Question 2 :
The angle between the pair of lines $y^{ 2 }\sin ^{ 2 }{ \theta <br>} -xy\ \sin ^{ 2 }{ \theta } +{ x }^{ 2 }\left( \cos ^{ 2 }{ <br>\theta } -1 \right) =0$ is <br>
Question 3 :
The angle between the pair of lines whose equation is $4 x ^ { 2 } + 10 x y + m y ^ { 2 } + 5 x + 10 y = 0$ is
Question 4 :
In a plane there are two families of lines y =x+r, y = -x+ r, where r ∈ {0, 1, 2, 3, 4}. The number of squares of diagonal of length 2 formed by these lines is/are greater than-
Question 5 :
The equation of the bisector of the angle between the lines $3x-4y+7=0$ and $12x+5y-2=0$
Question 6 :
Equation x<sup>2</sup> + k<sub>1</sub>y<sup>2</sup> + 2k<sub>2</sub>y = a<sup>2</sup> represents a pair of <br>perpendicular straight lines if
Question 7 :
The line joining $(-1,0) $ and $(-2, \displaystyle -\sqrt{3} )$ makes with the $x$-axis an angle equal to
Question 9 :
Let 2x -3y = 0 be given line and P (sin θ, 0) and Q (0, cosθ) be two points. Then P and Q lie on the same side of the given line if θ lies in the:
Question 10 :
The straight line $x+y-1=0$ meets the circle $x^2+y^2-6x-8y=0$ at A and B. Then the equation of the circle of which AB is a diameter is
Question 11 :
If one of the lines of <em>m</em><em>y</em><sup>2</sup> + (1−<em>m</em><sup>2</sup>)xy − <em>m</em><em>x</em><sup>2</sup> = 0 is a bisector of the angle between the lines xy = 0, then <em>m</em> is
Question 12 :
The lines x + y - 1 = 0, (m - 1) x + (m<sup>2 </sup>-7)y - 5 = 0 and (m - 2) x + (2m - 5) y = 0 are
Question 13 :
If $'\theta'$ is the angle between the lines $ax^2+2hxy+by^2=0$, then angle between $x^2+2xy sec\theta +y^2=0$ is
Question 14 :
If P,Q are two points on the line $3x+4y+15=0$ such that $OP=OQ=9$, then the area of $\triangle OPQ$ is ?
Question 15 :
If {tex} p {/tex} is the length of the perpendicular from the origin on the line {tex} \frac { x } { a } + \frac { y } { b } = 1 {/tex} and {tex} a ^ { 2 } , p ^ { 2 } , b ^ { 2 } {/tex} are in A.P. then {tex} a b {/tex} is equal to
Question 16 :
If the pair of lines represented by the equation {tex} 6 x ^ { 2 } + 17 x y + 12 y ^ { 2 } + 22 x + 31 y + 20 = 0 {/tex} be {tex} 2 x + 3 y + p = 0 {/tex} and {tex} 3 x + 4 y + q = 0 , {/tex} then
Question 17 :
The line making an angle $\left( -{ 120 }^{ o } \right) $ with $x$-axis is situated in the :
Question 18 :
The lines {tex} a x + ( b + c ) y = p , b x + ( c + a ) y = p {/tex} and {tex} c x {/tex} {tex} + ( a + b ) y = p {/tex} intersect at the point {tex} ( 3 / 4,3 / 4 ) {/tex} if
Question 20 :
The equation of a line through $(2, -3)$ parallel to y-axis is
Question 21 :
If the chord <em>y</em> = mx + 1 of the circle <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = 1 subtends an angle of measure 45<em></em><sup>∘</sup> at the major segment of the circle, then the value of <em>m</em> is
Question 22 :
If the two straight lines $\displaystyle y= m_{1}x + c_{1}$ and $\displaystyle y= m_{2}x + c_{2}$ are perpendicular to each other then $\displaystyle m_{1}m_{2}= $ _____________
Question 23 :
Two vertices of an equilateral triangle are {tex} ( - 1,0 ) , {/tex} {tex} ( 1,0 ) , {/tex} third vertex can be
Question 24 :
The angle between the pair of straight lines<br>$x^{ 2 }+4y^{ 2 }-7xy=0$ is
Question 25 :
One line passes through the points $(1,9)$ and $(2,6)$ another line passes through $(3, 3)$ and $(-1, 5)$ The acute angle between the two lines is
Question 26 :
If the angle between the two lines represented by $2x^2+5xy+3y^2+6x+7y+4=0$ is $\tan^{-1}$m then m is _______?
Question 27 :
What is the slope of the line 3x + 2y + 1 = 0?
Question 28 :
The equations of line AB and line PQ are y = $-\frac{1}{2}x$ and y=2x respectively. Find the measure of angle $\angle$ BOQ which is formed by intersection of line AB and line PQ. (Point P and point A are in first and second quadrant respectively)
Question 29 :
If the angle between two lines represented by $2x^{2} + 5xy + 3y^{2} + 7y + 4 = 0$ is $\tan^{-1}m$, then $m$ is equal to.
Question 30 :
$A(p, 0), B(4, 0), C(5, 6)$ and $D(1, 4)$ are the vertices of a quadrilateral $ABCD$. If $\angle ADC$ is obtuse, the maximum integral value of $p$ is :
Question 31 :
The combined equation of three sides of a triangle is (<em>x</em><sup>2</sup>−<em>y</em><sup>2</sup>)(2<em>x</em>+3<em>y</em>−6) = 0. If ( − 2, 0) is an interior point and (<em>b</em>, 1) is an exterior point of the triangle, then
Question 32 :
The acute angle between the lines<br>$l x + my =l+m, l(x-y) +m(x + y)=2m $ is <br>
Question 33 :
If the equations of the sides of a triangle are $x+y=0,$ and $\sqrt { 3 } y+x=0,$ then which of the following is an exterior point of the triangle?
Question 34 :
The lines $p\left( { p }^{ 2 }+1 \right) x-y+q=0$ and ${ \left( { p }^{ 2 }+1 \right) }^{ 2 }x+\left( { p }^{ 2 }+1 \right)y+2q=0$ are perpendicular to a common line for:<br><br>
Question 35 :
If the sum of the slopes of the lines given by $x^{2}-2cxy-7y^{2}= 0$ is four times their product then $c$ has the value
Question 36 :
The angle made by the line $\sqrt 3x-y+3=0$ with the positive direction of X-axis is
Question 37 :
The angle between line: $r[2 \cos \theta + 5\sin\theta ] = 3$ and $r[2 \sin \theta - 5\cos\theta ]+ 4 = 0$ is:<br/>
Question 38 :
Equation of the straight line parallel to $x + 2y -5 = 0$ and at the same distance from (3,2) is
Question 39 :
The straight lines $x + y = 0, 3x + y -4 = 0, x + 3y -4 = 0$ form a triangle which is
Question 40 :
If $A(-2,1),B(2,3)$ and $C(-2,-4)$ are three points, then the angle between $BA$ and $BC$ is:
Question 41 :
Tow consecutive sides of a parallelogram are $4x + 5y = 0$ and $7x + 2y = 0$.If the equation to one diagonal is $11x + 7y = 9$, then the equation of the other diagonal is <br>
Question 42 :
Two pairs of straight lines have the equations y<sup>2</sup> + xy -12x<sup>2</sup><br>= 0 and ax<sup>2</sup> + 2hxy + by<sup>2</sup> = 0.<br>One line will be common among them if
Question 43 :
Assertion: A chord of the curve $\displaystyle 3x^{2}-y^{2}-2x+4y=0$ passing through the point $(1, -2)$ subtend a right angle at the origin.
Reason: Lines represented by the equation $\displaystyle \left ( 3c+2m \right )x^{2}-2\left ( 1+2m \right )xy+\left ( 4-c \right )y^{2}=0$ are perpendicular if $c+m+2=0.$
Question 44 :
Which one of the following is correct in respect of the equations $\displaystyle\frac{x-1}{2}=\frac{y-2}{3}$ and $2x+3y=5$?
Question 45 :
If p < q and px<sup>2</sup> + 4u xy + qy<sup>2</sup> + 4a (x + y + 1) = 0 represents pair of st. line for some u ∈ R and a ≠ 0, then
Question 46 :
Find the point of intersection and the angle between the lines given by the equation : $2x^2+3xy-2y^2 + 10x + 5y + 12 = 0$<br>
Question 47 :
The angle between the lines $\sqrt { 3x } +y+1=0$ and $x+1=0$ is.
Question 48 :
If three parallel planes are given by<br>$P_1: 2x-y+2z=6$<br>$P_2: 4x-2y+4z=\lambda$<br>$P_3: 2x-y+2z=\mu$<br>If distance between $P_1$ and $P_2$ is $\dfrac{1}{3}$ and between $P_1$ and $P_3$ is $\dfrac{2}{3}$, then the maximum value of $\lambda +\mu$ is?<br>
Question 49 :
If the angle between the lines $ {k} {x}- {y}+6=0,\ 3 {x}-5 {y}+7=0$ is $\displaystyle \frac{\pi}{4},$ then one of the value of $ {k}=$ <br/>
Question 50 :
Assertion (A): The equation $2x^{2}+3xy-2y^{2}+5x-5y+3=0$ represents a pair of perpendicular straight lines.<br/>Reason (R): A pair of lines given by $ax^{2}+2hxy+by^{2}+2gx+2f +c=0$ are perpendicular, if $a+b=0$ <br/>
Question 51 :
The angle between the lines $ 2x+11y-7=0$<br>and $ x+3y+5=0$ is equal to :
Question 52 :
If the equation $6x^2 -11xy -10y^2 -2x-14y + c = 0$ represents a pair of lines, then which of the following shows the correct lines and the angle between them?
Question 53 :
Let {tex} L _ { 1 } {/tex} be a strainght line passing through the origin and {tex} L _ { 2 } {/tex} be the straight line {tex} x + y = 1 . {/tex} If the intercepts made by the circle {tex} x ^ { 2 } + y ^ { 2 } - x + 3 y = 0 {/tex} on {tex} L _ { 1 } {/tex} and {tex} L _ { 2 } {/tex} are equal, then which of the following equations can represent {tex} L _ { 1 } ? {/tex}
Question 54 :
The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal(measured counterclockwise from the positive x-axis) as does $L_2$ and that $L_1$ has $4$ times the slope of $L_2$. If $L_1$ is not horizontal, then the value of the product(mn) equals.
Question 55 :
If (<em>α</em>, <em>α</em><sup>2</sup>) lies inside the triangle formed by the lines 2<em>x</em> + 3<em>y</em> − 1 = 0, <em>x</em> + 2<em>y</em> − 3 = 0, 5<em>x</em> − 6<em>y</em> − 1 = 0, then
Question 56 :
If the equation ax<sup>2</sup> - 6xy + y<sup>2</sup> + bx + cy + d = 0 represents pair of lines whose slopes are m and m<sup>2</sup>, then value of a is/are
Question 57 :
The straight lines $4x-3y-5=0, x-2y-10=0, 7x+y-40=0$ and $x+3y+10=0$ form the sides of a<br/>
Question 58 :
If the sum of the slopes of the lines given by ${ x }^{ 2 }-2cxy-7{ y }^{ 2 }=0$ is four times their product, then $c$ has the value
Question 59 :
A straight line through the point A(-2, -3) cuts the line x + 3y = 9 and x + y + 1 = 0 at B and C respectively. If AB.AC = 20, then equation of the possible line is
Question 60 :
Measure of the angle between x + y = 0 and y = 5 is _______.
Question 61 :
If the line $\left( \cfrac { x }{ 2 } +\cfrac { y }{ 3 } -1 \right) +\lambda(2x+y-1)=0$ is parallel to x-axis then $\lambda=$
Question 62 :
If line $3 x + 4 y + c = 0$ lies midway between $3 x + 4 y + 10 = 0$ $\&$ $3 x + 4 y - 10 = 0$ then value of $c$ is
Question 64 :
If one of the diagonals of a square is along the line $x=2y$ and one of its verices is $(3,0)$, then its side through this vertex nearer to the origin is given by the equation.
Question 65 :
The angle between the lines $kx+y+9=0$, $y-3x =4$ is $45^{o},$ then the value of $k$ is :<br/>
Question 66 :
The angle between the lines $3x + y - 7 = 0$ and $x + 2y + 9 = 0$ is
Question 67 :
The <em>x</em>-coordintes of the vertices of a square of unit area are the roots of the equation <em>x</em><sup>2</sup> − 3|<em>x</em>| + 2 = 0 and the <em>y</em>-coordinates of the vertices are the roots of the equation <em>y</em><sup>2</sup> − 3<em>y</em> + 2 = 0. Then the possible vertices of the square is/are
Question 68 :
The equation $2x^2 + 4xy- py^2 + 4x + qy + 1 = 0$ will represent two mutually perpendicular straight lines, if<br>
Question 70 :
The line L given by $\dfrac { x }{ 5 } +\dfrac { y }{ b } =1$ passes through the point (13, 32). The line K is parallel to L and has the equation $\dfrac { x }{ e } +\dfrac { y }{ 3 } =1$. Then the distance between L and K is
Question 72 :
The angle between the line x+y=3 and the line joining the points (1,1) and (-3,4) is
Question 73 :
Let A be a variable point on the line y$=4$. If B and C are variable points on the line $y=-4$ such that triangle ABC is equilateral, then
Question 74 : The equation $x^2 \, - \, 5xy \, + \, py^2 \, + \, 3x \, - \, 8y \, + \, 2 \, = \, 0$ represent a pair of straigth lines. <div>If $\theta$ is the angle between them, then $\sin \theta \, = \, ?$</div>
Question 75 :
An angle $<XYZ=75^0$ is bisected by an angular bisector $YU$ , then the measure of $<UYZ$ is ______ .
Question 76 :
lf the equation $6\mathrm{x}^{2}+5\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{y}^{2}+9\mathrm{x}+20\mathrm{y}+\mathrm{c}=0$ represents a pair of perpendicular lines, then $\mathrm{b}-\mathrm{c}=$
Question 77 :
Find the acute angle between the lines $3x + y -7 = 0$ and $x + 2y- 9 = 0.$
Question 78 :
All points lying inside the triangle formed by the points {tex} ( 1,3 ) , ( 5,0 ) {/tex} and {tex} ( - 1,2 ) {/tex} satisfy
Question 79 :
The angle between the lines $\sin^2 \alpha \cdot y^2-2xy\cdot cos^2 \alpha + (\cos^2 \alpha -1)x^2=0$ is _______<br/>
Question 80 :
If one of the diagonals of a square is along the line $x=2y$ and one of its vertices is $\left(3,0\right)$, then its sides through this vertex are given by the equations
Question 81 :
The axes being inclined at an angle of 60$^o$, the angle between the two straight lines $y=2x+5$ and $2y +x + 7 =0$ is
Question 82 :
Lines $L_1 : x + \surd 3y = 2$, and $L_2 : ax + by = 1$ meet at $P$ and enclose an angle of $45^o$ between them. A line $L_3 : y = \surd 3x$, also passes through $P$ then<br>
Question 83 :
Assertion: If the angle between the two lines represented by $\displaystyle 2x^{2} + 5xy + 3y^{2} + 6x + 7y + 4 = 0$ is $\displaystyle \tan ^{-1} \left ( m \right )$, then value of $\displaystyle m = \frac{1}{5}$.
Reason: The angle $\displaystyle \theta$ between the two lines $\displaystyle ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ is calculated by $\displaystyle \tan^{-1} \frac{\left ( 2\sqrt{h^{2} - ab} \right )}{\left | a + b \right |}$.
Question 84 :
Let P be a point on the circle $x^2+y^2=9$ , Q a point on the line $7x + y + 3 = 0$, and the perpendicular bisector of PQ be the line $x-y + 1 = 0$. Then the coordinate of P can be
Question 85 :
If one of the diagonals of a square is along the line $4x=2y$ and one of its vertices is $(3,0),$ then its side through this vertex nearer to the origin is given by the equation.
Question 86 :
The condition on $a$ and $b,$ such that the portion of the line $ax+by- 1= 0$, intercepted between the lines $ax+y = 0$ and $x+by=0$, subtends a right angle at the origin is
Question 87 :
Consider $3$ lines as follows,<br>$L_1 : 5x - y + 4 = 0$<br>$L_2 : 3x - y + 5 = 0$<br>$L_3 : x + y + 8 = 0$<br>If these lines enclose a triangle $ABC$ and sum of the squares of the tangent to the interior angle can be expressed in the form $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime numbers, then the value of $p + q$ is
Question 88 :
The pair of lines $\sqrt { 3 } { x }^{ 2 }-4xy+\sqrt { 3 } { y }^{ 2 }=0$ are rotated about the origin by $\dfrac { \pi }{ 6 } $ in the anticlockwise sense. The equation of the pair in the new position is
Question 89 :
The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is :
Question 90 :
What is the angle between the straight lines $\left( { m }^{ 2 }-mn \right) y=\left( mn+{ n }^{ 2 } \right) x+{ n }^{ 3 }$ and $\left( mn+{ m }^{ 2 } \right) y=\left( mn-{ n }^{ 2 } \right) x+{ m }^{ 3 }$, where $m> n$?
Question 91 :
Find the angle of inclination (in degrees) of the line passing through the points $(1,2)$ and $(2,3)$.
Question 92 :
The straight lines represented by $(y - mx)^2 = a^2 (1+m^2)$ and $(y-nx)^2 =a^2 (1 +n^2)$ form a
Question 93 :
If $(a, b)$ is the co-ordinates of the point obtained in previous question, then the equation of line which is at the distance $|b - 2a - 1|$ units from origin and make equal intercept on co-ordinate axes in first quadrant, is
Question 94 :
The axes being inclined at an angle of $60^{\circ}$, the inclination of the straight line $y = 2x + 5$ with x-axis is
Question 95 :
One diagonal of a square is along the line $8x - 15 y =0$ and one of its vertices is $(1, 2)$. Then the equations of the sides of the square passing through this vertex are
Question 96 :
If $\theta$ is angle between two non parallel lines then $\sin \theta$ is
Question 97 :
The angle made by the line joining the points $(2,0)$ and $( - 4,2\sqrt 3 )$ with x axis is -
Question 98 : <p>Two sides of a triangle are the lines (<em>a</em>+<em>b</em>)<em>x</em> + (<em>a</em>−<em>b</em>)<em>y</em> − 2ab = 0</p> <p>(<em>a</em>−<em>b</em>)<em>x</em> + (<em>a</em>+<em>b</em>)<em>y</em> − 2ab = 0. If the triangle is isosceles and the third side passes through point (<em>b</em> − <em>a</em>, <em>a</em> − <em>b</em>), then the equation of third side can be</p>
Question 99 :
A straight line passes through the points of intersection $x-2y-2=9$ and $2x-bx-6=0$ and the origin. Then the complete set of values ob $b$ for which the acute angle between this line and $y=0$ is less $45^{o}$ is
Question 100 :
A line $4x+y=1$ passes through the point $A(2,-7)$ and meets line $BC$ at $B$ whose equation is $3x-4y+1=0,$ the equation of line $AC$ such that $AB=AC$ is
Question 101 :
The diagonals of a parallelogram $PQRS$ are along the lines $x+3y=4$ and $6x-2y=7,$ then $PQRS$ must be a
