Pair of straight lines practice · Mixed difficulty

If the bisectors of the lines $x^2-2 p x y-y^2=0$ be $x^2-2 q x y-y^2=0$, then

If the equation $12\;{x^2} + 7\;xy - p{y^2} - 18\;x + qy + 6 = 0$ represents a pair of perpendicular straight lines, then

If the equation $\lambda x^2+2 y^2-5 x y+5 x-7 y+3=0$ represents two straight lines, then find the value of $\lambda$

If the pair of lines $a x^2-2 x y+b y^2=0$ and $b x^2-2 x y+a y^2=0$ be such that each pair bisects the angle between the other pair, then $|a-b|$ equals to

If the equation $4 x^2+4 x y+y^2-6 x-3 y-4=0$ represents a pair of parallel line, then the distance between these lines is

Area of the triangle formed by the lines $3{x^2} - 4xy + {y^2} = 0,\;2x - y = 6$ is

Equation of a straight line passing through the point of intersection of $x-y+1=0$ and $3 x+y-5=0$ are perpendicular to one of them is

The lines represents by $a{x^2} + 2hxy + b{y^2} = 0$ are perpendicular to each other, if

The sides of a rectangle are $x=0, y=0, x=4$ and $y=3$. The equation of the straight line having slope $\frac{1}{2}$ that divides the rectangle into two equal halves, is

The set of values of $h$ for which the equation $4 x^2+h x y-3 y^2=0$ represents a pair of real and distinct lines is

The four straight lines given by the equations $12 x^2+7 x y-12 y^2=0$ and
$12 x^2+7 x y-12 y^2-x+7 y-1=0$ lie along the sides of a

If the segment of the line between the lines $x-y+2=0$ and $x+y+4=0$ is bisected at the origin, equation of the line is

Perimeter of the quadrilateral bounded by the coordinate axis and the lines $x+y=50$ and $3 x+y=90$ is

Pair of straight lines perpendicular to each other represented by

If the lines $3 x-y+1=0$ and $x-2 y+3$ $=0$ are equally inclined to the line $y=m x$, then the value of $m$ is given by

$P$ is a point on $x$-axis, $Q$ is a point on $y$-axis. Both are equidistant from the points $(7,6)$ and $(3,4)$. Distance between $P$ and $Q$ is

Distance between the parallel lines $4 x^2+20 x y+25 y^2+2 x+5 y-12=0$ is

If two pairs of straight lines having equations $y^2+x y-12 x^2=0$ and $a x^2+2 h x y+b y^2=0$ have one line common, then $a=$

If the lengths of the perpendiculars drawn from origin to $x \sec \theta+y \operatorname{cosec} \theta=a$ and $x \cos \theta-y \sin \theta=a \cos 2 \theta$ are $p$ and $p^{\prime}$ respectively, then $4 p^2+p^{\prime 2}=\ldots \ldots$

Equations of the bisectors of the lines $3 x-4 y+7=0$ and $12 x+5 y-2=0$ are given by

The distance between the pair of lines represented by the equation ${x^2} - 6xy + 9{y^2} + 3x - 9y - 4 = 0$ is

The combined equation of the pair of lines through the origin and perpendicular to the pair of lines given by $a x^2+2 h x y+b y^2=0$, is

Separate equations of lines for a pair of lines whose equation is ${x^2} + xy - 12{y^2} = 0$, are

If the equation of the pair of straight lines passing through the point $(1,1)$, one making an angle $\theta$ with the positive direction of the $X$-axis and the other making the same angle with the positive direction of the $Y$-axis, is $x^2-(a+2) x y+y^2+a(x+y-1)=0, a \neq-2$, then the value of $\sin 2 \theta$ is

The distance between the pair of parallel lines ${x^2} + 4xy + 4{y^2} + 3x + 6y - 4 = 0$ is

If the equation $x^2+(\lambda+\mu) x y+\lambda \mu y^2+x+\mu y=0$ represents two parallel straight lines, then

If $a{x^2} - {y^2} + 4x - y = 0$ represents a pair of lines, then $a$ is equal to

The difference of the tangents of the angles which the lines $x^2\left(\sec ^2-\sin ^2 \theta\right)-2 x y \tan \theta+y^2 \sin ^2 \theta=0$ make with $\mathrm{X}$-axis, is

The lines joining the points of intersection of line $x+y=1$ and curve $x^2+y^2-2 y+\lambda=0$ to the origin are perpendicular, then the value of $\lambda$ will be

If the slope of one line in the pair $a x^2+4 x y+y^2=0$ is three times the other, then $a=$