Straight Lines and Pair of Straight Lines · Mathematics · MHT CET

The co-ordinates of the foot of perpendicular, drawn from the point $(-2,3)$ on the line $3 x-y-1=0$ are

If $P_1$ and $P_2$ are perpendicular distances (in units) from point $(2,-1)$ to the pair of lines $2 x^2-5 x y+2 y^2=0$, then the value of $\mathrm{P}_1 \mathrm{P}_2$ is

If $\frac{x^2}{\mathrm{a}}+\frac{2 x y}{\mathrm{~h}}+\frac{y^2}{\mathrm{~b}}=0$ represents a pair of straight lines and slope of one of the lines is twice that of the other, then $a b: h^2$ is

Suppose that the points $(h, k),(1,2)$ and $(-3,4)$ lie on the line $l_1$. If a line $l_2$ passing through the points $(h, k)$ and $(4,3)$ is perpendicular to $l_1$, then $\left(\frac{k}{h}\right)$ equals

If one of the lines represented by $a x^2+2 h x y+b y^2=0$ is perpendicular to $\mathrm{m} x+\mathrm{n} y=18$, then

If two sides of a square are $4 x+3 y-20=0$ and $4 x+3 y+15=0$, then the area of the square is

The value of $k$, if the slope of one of the lines given by $4 x^2+k x y+y^2=0$ is four times that of the other, is given by

The joint equation of pair of lines through the origin, each of which makes an angle of $30^{\circ}$ with Y -axis, is

A straight line L through the point $(3,-2)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} x+y=1$. If L also intersects the X -axis, then the equation of $L$ is

The joint equation of two lines through the origin, each making an angle with measure of $30^{\circ}$ with the positive Y -axis, is

If the slope of one of the lines given by $\mathrm{K} x^2+6 x y+y^2=0$ is three times the order, then the value of $K$ is

Let $\mathrm{P} \equiv(-5,0), \mathrm{Q} \equiv(0,0)$ and $\mathrm{R} \equiv(2,2 \sqrt{3})$ be three points. Then the equation of the bisector of the angle $P Q R$ is

If the length of the perpendicular to a line from the origin is $2 \sqrt{2}$ units, which makes an angle of $135^{\circ}$ with the X -axis, then the equation of line is

The number of integer values of $m$, for which $x$-coordinate of the point of intersection of the lines $3 x+4 y=9$ and $y=m x+1$ is also an integer, is

Let a line intersect the co-ordinate axes in points $A$ and $B$ such that the area of the triangle $O A B$ is 12 sq. units. If the line passes through the point $(2,3)$, then the equation of the line is

$\triangle \mathrm{OAB}$ is formed by the lines $x^2-4 x y+y^2=0$ and the line $A B$. The equation of line $A B$ is $2 x+3 y-1=0$. Then the equation of the median of the triangle drawn from the origin is

The joint equation of pair of lines through the origin and making an angle of $\frac{\pi}{6}$ with the line $3 x+y-6=0$ is

A line $4 x+y=1$ passes through the point $\mathrm{A}(2,-7)$ meets the line BC whose equation is $3 x-4 y+1=0$ at the point $B$. The equation of the line $A C$ so that $A B=A C$ is

If an equation $h x y+g x+f y+c=0$ represents a pair of lines, then

The straight line, $2 x-3 y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15, \beta)$, then $\beta$ equals

If $\mathrm{O}(0,0), \mathrm{A}(1,2)$ and $\mathrm{B}(3,4)$ are the vertices of triangle OAB , then the joint equation of the altitude and median drawn from O is

The combined equation of two lines through the origin and making an angle of $45^{\circ}$ with the line $3 x+y=0$, is

The acute angle between the lines $x \cos 30^{\circ}+y \sin 30^{\circ}=3$ and $x \cos 60^{\circ}+y \sin 60^{\circ}=5$ is

If $4 a b=3 h^2$, then the ratio of the slope of lines represented by $a x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0$ is

The line L given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line K is parallel to line L and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between L and K is _________ units.

The equation of the line passing through the point of intersection of the lines $3 x-y=5$ and $x+3 y=1$ and making equal intercepts on the axes is

The equation of pair of lines $y=p x$ and $y=q x$ can be written as $(y-p x)(y-q x)=0$. Then the equation of the pair of the angle bisectors of the lines $x^2-4 x y-5 y^2=0$ is

The number of possible distinct straight lines passing through $(2,3)$ and forming a triangle with co-ordinate axes whose area is 12 sq . units are,

The line L given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line K is parallel to L and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

The diagonals of a parallelogram $A B C D$ are along the lines $x+3 y=4$ and $6 x-2 y=7$. Then ABCD must be a

If the equation $7 x^2-14 x y+p y^2-12 x+q y-4=0$ represents a pair of parallel lines then the value of $\sqrt{p^2+q^2-p q}$ is

The slopes of the lines given by $x^2+2 h x y+2 y^2=0$ are in the ratio $1: 2$, then $h$ is

If two lines $x+(a-1) y=1 \quad$ and $2 x+a^2 y=1(a \in R-\{0,1\})$ are perpendicular, then the distance of their point of intersection from the origin is

Let $$P \equiv(-3,0), Q \equiv(0,0)$$ and $$R \equiv(3,3 \sqrt{3})$$ be three points. Then the equation of the bisector of the angle $$\mathrm{PQR}$$ is

The centroid of the triangle formed by the lines $$x+3 y=10$$ and $$6 x^2+x y-y^2=0$$ is

$$\mathrm{p}$$ is the length of perpendicular from the origin to the line whose intercepts on the axes are a and $$\mathrm{b}$$ respectively, then $$\frac{1}{\mathrm{a}^2}+\frac{1}{\mathrm{~b}^2}$$ equals

The perpendiculars are drawn to lines $$L_1$$ and $$L_2$$ from the origin making an angle $$\frac{\pi}{4}$$ and $$\frac{3 \pi}{4}$$ respectively with positive direction of $$\mathrm{X}$$-axis. If both the lines are at unit distance from the origin, then their joint equation is

Let $$P Q R$$ be a right angled isosceles triangle, right angled at $$Q(2,1)$$. If the equation of the line $$P R$$ is $$2 x+y=3$$, then the combined equation representing the pair of lines $$P Q$$ and $$Q R$$ is

$$P S$$ is the median of the triangle with vertices at $$P(2,2), Q(6,-1)$$ and $$R(7,3)$$, then the intercepts on the coordinate axes of the line passing through point $$(1,-1)$$ and parallel to PS are respectively

If the angle between the lines given by $$x^2-3 x y+\lambda y^2+3 x-5 y+2=0 ; \lambda \geq 0$$ is $$\tan ^{-1}\left(\frac{1}{3}\right)$$, then the value of $$\lambda$$ is

The base of an equilateral triangle is represented by the equation $$2 x-y-1=0$$ and its vertex is $$(1,2)$$, then the length (in units) of the side of the triangle is

A line is drawn through the point $$(1,2)$$ to meet the co-ordinate axes at $$\mathrm{P}$$ and $$\mathrm{Q}$$ such that it forms a $$\triangle \mathrm{OPQ}$$, where $$\mathrm{O}$$ is the origin. If the area of $$\triangle \mathrm{OPQ}$$ is least, then the slope of the line $$\mathrm{PQ}$$ is

If the pair of lines given by $$(x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2\right) \sin ^2 \alpha$$ are perpendicular to each other, then $$\alpha$$ is

If $$\mathrm{k}_{\mathrm{i}}$$ are possible values of $$\mathrm{k}$$ for which lines $$\mathrm{k} x+2 y+2=0,2 x+\mathrm{k} y+3=0$$ and $$3 x+3 y+\mathrm{k}=0$$ are concurrent, then $$\sum \mathrm{k}_{\mathrm{i}}$$ has the value

The co-ordinates of the points on the line $$2 x-y=5$$ which are the distance of 1 unit from the line $$3 x+4 y=5$$ are

Let $$\mathrm{PQR}$$ be a right angled isosceles triangle, right angled at $$\mathrm{P}(2,1)$$. If the equation of the line $$\mathrm{QR}$$ is $$2 x+y=3$$, then the equation representing the pair of lines $$P Q$$ and $$P R$$ is

The joint equation of the lines pair of lines passing through the point $$(3,-2)$$ and perpendicular to the lines $$5 x^2+2 x y-3 y^2=0$$ is

If the angle between the lines represented by the equation $$x^2+\lambda x y-y^2 \tan ^2 \theta=0$$ is $$2 \theta$$, then the value of $$\lambda$$ is

$$\mathrm{a}$$ and $$\mathrm{b}$$ are the intercepts made by a line on the co-ordinate axes. If $$3 \mathrm{a}=\mathrm{b}$$ and the line passes through $$(1,3)$$, then the equation of the line is

The joint equation of a pair of lines passing through the origin and making an angle of $$\frac{\pi}{4}$$ with the line $$3 x+2 y-8=0$$ is

Two sides of a square are along the lines $$5 x-12 y+39=0$$ and $$5 x-12 y+78=0$$, then area of the square is

The number of integral values of $$\mathrm{p}$$ in the domain $$[-5,5]$$, such that the equation $$2 x^2+4 x y-p y^2+4 x+q y+1=0$$ represents pair of lines, are

The points $$(1,3),(5,1)$$ are opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line $$y=2 x+\mathrm{c}$$, then one of the vertex on the other diagonal is

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

The equation of a line, whose perpendicular distance from the origin is 7 units and the angle, which the perpendicular to the line from the origin makes, is $$120^{\circ}$$ with positive $$\mathrm{X}$$-axis, is

If the distance between the parallel lines given by the equation $$x^2+4 x y+p y^2+3 x+q y-4=0$$ is $$\lambda$$, then $$\lambda^2=$$

The distance of a point $$(2,5)$$ from the line $$3 x+y+4=0$$ measured along the line $$L_1$$ and $$L_1$$ are same. If slope of line $$L_1$$ is $$\frac{3}{4}$$, then slope of the line $$\mathrm{L}_2$$ is

The joint equation of pair of lines through the origin and making an equilateral triangle with the line $$y = 5$$ is

The equations of the lines passing through the point $$(3,2)$$ and making an acute angle of $$45^{\circ}$$ with the line $$x-2 y-3=0$$ are

If the polar co-ordinates of a point are $$\left(2, \frac{\pi^{\mathrm{c}}}{4}\right)$$, then its Cartesian co-ordinates are

The acute angle between the lines $$\left(x^2+y^2\right) \sin \theta+2 x y=0$$ is

If the lines represented by $$(k^2+2) x^2+3 x y-6 y^2=0$$ are perpendicular to each other, then the values of $$\mathrm{K}$$ are

The equation of a line with slope $$-\frac{1}{\sqrt{2}}$$ and makes an intercept of $$2 \sqrt{2}$$ units on negative direction of $$y$$-axis is

If the lines respresented by $$a x^2-b x y-y^2=0$$ make angle $$\alpha$$ and $$\beta$$ with the positive direction of $$\mathrm{X}$$-axis, then $$\tan (\alpha+\beta)=$$

If the angle between the lines is $$\frac{\pi^{\mathrm{C}}}{4}$$ and slope of one of the lines is $$\frac{1}{2}$$, then slope of the other line is

If one of the lines given by $$k x^2+x y-y^2=0$$ bisect the angle between the co-ordinate axes then the values of $$k$$ are

The joint equation of pair of lines through the origin and having slopes $$(1+\sqrt{2})$$ and $$\frac{1}{(1+\sqrt{2})}$$ is

If $$4 a b=3 h^2$$, then the ratio of slopes of the lines represented by $$a x^2+2 h x y+b y^2=0$$ is

The distance between the lines $$3 x+4 y=9$$ and $$6 x+8 y=15$$ is

If the slopes of the lines given by the equation $$a x^2+2 h x y+b y^2=0$$ are in the ratio $$5: 3$$, then the ratio $$h^2: a b=$$

The equation of line, where length of the perpendicular segment from origin to the line is 4 and the inclination of this perpendicular segment with the positive direction of X-axis is 30$$^\circ$$, is

If two lines represented by $$a x^2+2 h x y+b y^2=0$$ makes angles $$\alpha$$ and $$\beta$$ with positive direction of $$\mathrm{X}$$-axis, then $$\tan (\alpha+\beta)=$$

The combined equation of a pair of lines passing through the origin and inclined at $$60^{\circ}$$ and $$30=$$ respectively with $$x$$-axis is

If the sum of slopes of lines represented by $$\mathrm{ax^2+8xy+5y^2=0}$$ is twice their product, then a =

If the line joining two points $$\mathrm{A}(2,0)$$ and $$\mathrm{B}(3,1)$$ is rotated about $$\mathrm{A}$$ in anticlockwise direction through an angle of $$15^{\circ}$$, then the equation of the line in new position is

If lines represented by the equation $$\mathrm{px}^2-\mathrm{qy^{2 }}=0$$ are distinct, then

If slope of one of the lines ax$$^2$$ + 2hxy + by$$^2$$ = 0 is twice that of the other, then h$$^2$$ : ab is

Area of the triangle formed by the lines $$y^2-9 x y+18 x^2=0$$ and $$y=9$$ is

The equation of perpendicular bisector of the line segment joining $$A(-2,3)$$ and $$B(6,-5)$$ is

If $$\mathrm{(m+3 n)(3 m+n)=4 h^2}$$, then the acute angle between the lines represented by $$\mathrm{m x^2+2 h x y+n y^2=0}$$ is

If $$\mathrm{p}$$ is the length of the perpendicular from origin to the line whose intercepts on the axes are a and $$b$$, then $$\frac{1}{a^2}+\frac{1}{b^2}=$$

If the lines $$x^2-4xy+y^2=0$$ make angles $$\alpha$$ and $$\beta$$ with positive direction X-axis, then $$\cot^2\alpha+\cot^2\beta=$$

If the two lines given by $$a x^2+2 h x y+b y^2=0$$ make inclinations $$\propto$$ and $$\beta$$, then $$\tan (\alpha+\beta)=$$

If the polar co-ordinates of a point are $$\left(\sqrt{2}, \frac{\pi}{4}\right)$$, then its Cartesian co-ordinates are

The equation of a line passing through $$(\mathrm{p} \cos \propto, \mathrm{p} \sin \propto)$$ ) and making an angle $$(90+\propto)$$ with positive direction of $$\mathrm{X}$$-axis is

The product of the perpendicular distances from $$(2,-1)$$ to the pair of lines $$2 x^2-5 x y+2 y^2=0$$ is

The x-intercept of a line passing through the points $$\left(\frac{-1}{2}, 1\right)$$ and (1, 2) is :

If the equation $$3x^2-kxy-3y^2=0$$ represents the bisectors of angles between the lines $$x^2-3xy-4y^2=0$$, then value of k is

The joint equation of pair of lines through the origin and making an equilateral triangle with the line $$y=3$$ is

If the acute angle between the lines given by $$\mathrm{a x^2+2 h x y+b y^2=0}$$ is $$\frac{\pi}{4}$$, then $$\mathrm{4 h^2=}$$

The joint equation of the pair of lines through the origin and making an equilateral triangle with the line $$x=3$$ is

The slope of the line through the origin which makes an angle of 30$$^\circ$$ with the positive direction of Y-axis measured anticlockwise is :

If the slopes of the lines given by the equation $a x^2+2 h x y+b y^2=0$ are in the ratio $5: 3$, then the ratio $h^2: a b=$

The equation of a line passing through the point $(7,-4)$ and perpendicular to the line passing through the points $(2,3)$ and $(1,-2)$ is

If the equation $3 x^2+10 x y+3 y^2+16 y+k=0$ represents a pair of lines, then the value of kis

If the equation $$a x^2+2 h x y+b y^2+2 g x+2 f y=0$$ has one line as the bisector of the angle between co-ordinate axes, then

The straight lines represented by the equation $$9 x^2-12 x y+4 y^2=0$$ are

If the equation $$k x y+5 x+3 y+2=0$$ represents a pair of lines, then $$k=$$

If $$(a,-2 a), a>0$$ is the mid-point of a line segment intercepted between the co-ordinate axes, then the equation of the line is

If the angle between the lines given by the equation $$x^2-3 x y+\lambda y^2+3 x-5 y+2=0, \lambda \geq 0$$, is $$\tan ^{-1}\left(\frac{1}{3}\right)$$, then $$\lambda=$$

The joint equation of lines passing through origin and having slopes $(1+\sqrt{2})$ and $\frac{-1}{1+\sqrt{2}}$ is ..........

The polar co-ordinates of $P$ are $\left(2, \frac{\pi}{6}\right)$. If $Q$ is the image of $P$ about the $X$-axis then the polar co-ordinates of $Q$ are.....̣...

The acute angle between lines $x-3=0$ and $x+y=19$ is.......

If sum of the slopes of the lines given by $x^2-4 p x y+8 y^2=0$ is three times their product then $p=$ ...........

If lines represented by $$\left(1+\sin ^2 \theta\right) x^2+2 h x y+2 \sin \theta y^2=0, \theta \in[0,2 \pi]$$ are perpendicular to each other then $\theta=$ ...........

If $(-\sqrt{2}, \sqrt{2})$ are cartesian co-ordinates of the point, then its polar co-ordinates are .........

The $y$-intercept of the line passing through $A(6,1)$ and perpendicular to the line $x-2 y=4$ is ...........

The joint equation of pair of straight lines passing through origin and having slopes $(1+\sqrt{2})$ and $\left(\frac{1}{1+\sqrt{2}}\right)$ is .......

The joint equation of the lines passing through the origin and trisecting the first quadrant is

If $P(2,2), Q(-2,4)$ and $R(3,4)$ are the vertices of $\triangle P Q R$ then the equation of the median through vertex $R$ is ......