Pair of straight lines practice · Mixed difficulty

A line passing through the point $P(2,3)$ meets the lines represented by $x^2-2 x y-y^2=0$ at the points $A$ and $B$ such that $P A \cdot P B=17$, the equation of the line is

A line which makes an acute angle $\theta$ with the positive direction of $x$-axis is drawn through the point $P$ $(3,4)$ to meet the lines $x=6$ and $y=8$ at $R$ and $S$ respectively then $R S=8-2 \sqrt{3}$, if

The angle between the pair of straight lines $x^2+4 y^2-7 x y=0$, is

The straight lines represented by $x^2+m x y-2 y^2+3 y-1=0$ meet at the point

The straight lines $7 x-2 y+10=0$ and $7 x+2 y-10=0$ forms and isosceles triangle with the line $y=2$, area of this triangle is equal to

The point of intersection of the two lines given by $2 x^2-5 x y+2 y^2+3 x+3 y+1=0$ is

The line $y=m x$ bisects the angle between the lines $a x^2+2 h x y+b y^2=0,$ if

The angle between the pair of straight lines $y^2 \sin ^2 \theta-x y \sin ^2 \theta+x^2\left(\cos ^2 \theta-1\right)=0$, is

The value of 'a' for which the lines represented by $a{x^2} + 5\;xy + 2\;{y^2} = 0$ are mutually perpendicular is

If the line represented by ${x^2} - 2\;pxy - {y^2} = 0$ are rotated about the origin through an angle $\theta$, one in clockwise direction and other in anti-clockwise direction, then the equation of the bisectors of the angle between the lines in the new position is

The distance between the parallel lines $9{x^2} - 6xy + {y^2} + 18x - 6y + 8 = 0,$ is

If the lines $x=k ; k=1,2 \ldots, n$ meet the line $y=3 x+4$ at the points $A_k\left(x_k, y_k\right), k=1,2, \ldots, n$ then the ordinate of the centre of Mean position of the points $A_k$, $k=1,2, \ldots, n$ is

If the slope of one of the lines represented by $a x^2+$ $(3 a+1) x y+3 y^2=0$ be reciprocal of the slope of the other, then the slopes of the lines are

The equation of the lines which are passing through the origin and which are perpendicular to the pair of straight lines $x^2-3 x y+2 y^2=0$ is

The equation $3 a x^2+9 x y+\left(a^2-2\right) y^2=0$ represents two perpendicular straight lines for

The angle between the pair of lines given by $3 x^2+5 x y-2 y^2+x+9 y-4=0$

The equation $x^3+y^3=0$ represents

The equation of the line whose perpendicular distance from the origin is 3 units and the angle which the normal makes with the positive direction of $x$-axis is $30^{\circ}$ is

The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented by $2{x^2} - xy - {y^2} = 0$ is

If $\theta$ is the acute angle between the lines given by $x^2-2 p x y+y^2=0$, then

The angle between the lines ${x^2} + 4xy + {y^2} = 0$ is

The area of the triangle formed by the lines $4 x^2-9 x y-9 y^2=0$ and $x=2$ is equal to

If the pairs of lines $x^2+2 x y+\lambda y^2=0$ and $\lambda x^2+2 x y+y^2=0$ have exactly one line in common, then the joint equation of the other two lines is given by

If the distance between the parallel lines $3 x+4 y+7=0$ and $a x+y+b=0$ is 1 , the integral value of $b$ is

If the straight lines $a x^2+2 h x y+b y^2$ $+2 g x+2 f y+c=0$ intersect on the $X$-axis, then

The equation of the bisector of the acute angle between the line $3x - 4y + 7 = 0$ and $12x + 5y - 2 = 0$ is

Two lines represented by the equation $x^2-y^2-2 x+1=0$ are rotated about the point $(1,0)$, the line making the bigger angle with the positive direction of the $x$-axis being turned by $45^{\circ}$ in the clockwise sense and the other line being turned by $15^{\circ}$ in the anticlockwise sense. The combined equation of the pair of lines in their new position, is

The angle $\theta$ between the pairs of straight lines represented by $6 x^2-x y-12 y^2-8 x+29 y-14=0$ is such that $\tan \theta$ equals

The equation $x^2-3 x y+\lambda y^2+3 x-5 y+2=0$ when $\lambda$ is a real number, represents a pair of straight lines. If $\theta$ is the angle between the lines, then $\operatorname{cosec}^2 \theta=$

If the pairs of straight lines $a x^2+2 h x y-a y^2=0$ and $b x^2+2 g x y-b y^2=0$ be such that each bisects the angles between the other, then