Straight Lines and Pair of Straight Lines PYQs - Last 10 Years

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Last 10 Years Straight Lines and Pair of Straight Lines Questions

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1Straight Lines And Pair Of Straight Lines

The number of real values of $\alpha$ for which the pair of lines represented by $\left(\alpha^2+12|\alpha|\right) x^2+6 x y+(18-21|\alpha|) y^2=0$ are at right angles to each other, is

2Straight Lines And Pair Of Straight Lines

If the lines $L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0$ and $L_3 \equiv 3 x+y+c=0$ are concurrent and $\theta$ is the acute angle between the lines $L_1=0$ and $L_3=0$, then $\tan \theta=$

3Straight Lines And Pair Of Straight Lines

If $3 x+6 y+2=0, x+y+1=0,2 x-y+3=0$ are three given lines, then the point $\left(\frac{-4}{3}, \frac{1}{3}\right)$ is

4Straight Lines And Pair Of Straight Lines

If the lengths of the perpendiculars drawn from a point $(a, b)$ to the lines $2 x+3 y+4=0$ and $3 x-2 y+4=0$ are same, then the point $(a, b)$ lies on the line

5Straight Lines And Pair Of Straight Lines

The line $x+2 y-c=0$ meets the curve $x^2+y^2-3 x-6 y+3=0$ at two points $P$ and $Q$ and $\angle P O Q=\frac{\pi}{2}$, where $O$ is the origin. Then, $2 c^2-15 c=$

6Straight Lines And Pair Of Straight Lines

If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$

7Straight Lines And Pair Of Straight Lines

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

8Straight Lines And Pair Of Straight Lines

Let the line $L$ drawn perpendicular to the lines $2 x-3 y+4=0$ and $6 x-9 y+7=0$ meet them at $A$ and $B$, respectively. If $P(\mathrm{l}, \mathrm{l})$ is a point on $L$, then the ratio in which $P$ divides $A B$ is

9Straight Lines And Pair Of Straight Lines

The orthocentre of the triangle formed by the points $(1,3),(-3,5)$ and $(5,-1)$ is

10Straight Lines And Pair Of Straight Lines

If the line $2 x-y-4=0$ divides the line segment joining the points $(2,-1)$ and $(1,-4)$ at the point $(a, b)$ in the ratio $m: n$, then $4\left(a-b\left(\frac{m}{n}\right)^2\right)=$

11Straight Lines And Pair Of Straight Lines

If the lines joining the origin to the points of intersection of the line $x+y=k$ and the curve $x^2+y^2-2 x-4 y+2=0$ are at right angles, then the sum of all the possible values of $k$ is

12Straight Lines And Pair Of Straight Lines

$\frac{x^2}{a}+\frac{x y}{h}+\frac{y^2}{b}=0(a \neq 0, h \neq 0, b \neq 0)$ represents two coincident lines if

13Straight Lines And Pair Of Straight Lines

If $\alpha x^2+2 \gamma x y+\beta y^2=0$ is the equation of pair of lines passing through the origin and perpendicular to the pair of lines $b h x^2+a b x y+a h y^2=0(a \neq 0, b \neq 0)$, then $\alpha \beta / \gamma^2=$

14Straight Lines And Pair Of Straight Lines

The distance between the points of concurrency of the two families of straight lines given by $x+(5 \lambda+1) y+1-3 \lambda=0$ and $(5 \mu+2) x-3 y+3+6 \mu=0$ is

15Straight Lines And Pair Of Straight Lines

If $f(x, y)=0$ is the combined equation of the lines joining the origin to the points where the line $4 x-6 y-2=0$ meets the curve $3 x^2-4 x y+5 y^2-2 x+y-6=0$, then $\frac{f(1,-1)}{f(-1,-1)}=$

16Straight Lines And Pair Of Straight Lines

If $\theta$ is the acute angle between the pair of lines $H \equiv a x^2-x y+b y^2=0, \tan \theta=5$ and $(1,-1)$ is a point on $H=0$, then $a^2+a b+b^2=$

17Straight Lines And Pair Of Straight Lines

If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable line is zero, then the variable line always passes through a fixed point. The coordinates of that point are

18Straight Lines And Pair Of Straight Lines

If a straight line $L$ passing through the point $(5,-3)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} x+y-9=0$ and $L$ intersects $X$-axis, then the equation of $L$ is

19Straight Lines And Pair Of Straight Lines

For $a, b, c \in R$, if $6 a^2-3 b^2-c^2+7 a b-a c+4 b c=0$ and $|a|+|b| \neq 0$, then all the lines given by $a x+b y+c=0$ are

20Straight Lines And Pair Of Straight Lines

Let $\alpha, \beta$ and $\gamma$ be three non-zero real constants and $a, b$ and $c$ be three arbitrary real numbers which satisfy $\alpha a+\beta b+\gamma c=0$. Then, the point of concurrence of the family of lines $a x+b y+c=0$ is

21Straight Lines And Pair Of Straight Lines

The equation of the pair of straight lines passing through the point $(2,3)$ and perpendicular to the pair of lines $3 x^2-4 x y+5 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, then $a+b+c+f+g+h=$

22Straight Lines And Pair Of Straight Lines

Let $L_1, L_2$ be the lines represented by the equation $4 x^2-5 x y+3 y^2=0$. Let $L_3, L_4$ be two lines passing through the point $(4,3)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equati...

23Straight Lines And Pair Of Straight Lines

If the incentre and the circumcentre of the triangle formed by the lines $x=2,4 x+3 y+7=0$ and $y=3$ are $I$ and $S$ respectively, then $I S=$

24Straight Lines And Pair Of Straight Lines

The equation $x^2-y^2+a x+b=0$ represents a pair of lines for the ordered pair $(a, b)=$

25Straight Lines And Pair Of Straight Lines

$a x^2-4 x y-2 y^2=0$ represents a pair of lines. If $\theta$ is the angle between these lines, $\cos \theta=\frac{1}{5}$ and the possible values of ' $a$ ' are $a_1$ and $a_2\left(a_1

26Straight Lines And Pair Of Straight Lines

Let $A(5,-3), B(3,-2), C(-1,5)$ be three points. If $P$ is a point satisfying the condition $P A^2+2 P B^2=3 P C^2$, then a point that lies on the locus of $P$ is

27Straight Lines And Pair Of Straight Lines

If the line $x-y+1=0$ cuts the lines $2 x+2 y+3=0$ and $3 x+3 y+2=0$ at the points $A$ and $B$ respectively, then $A B=$

28Straight Lines And Pair Of Straight Lines

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1$, then $\sin \theta=$

29Straight Lines And Pair Of Straight Lines

If $Q$ is the image of the point $P(1,1)$ with respect to the straight line $x+y+1=0$, then the length of the perpendicular drawn from $Q$ to the line $3 x-4 y+3=0$ is

30Straight Lines And Pair Of Straight Lines

If the lines $L_1 \equiv 2 x+y+3=0, L_2 \equiv k x+2 y-3=0$ and $L_3 \equiv 3 x-2 y+1=0$ are concurrent then the cosine of the acute angle between the lines $L_2=0$ and $2 x-5 y+7=0$ is

31Straight Lines And Pair Of Straight Lines

If the line $2 x-3 y+4=0$ divides the line segment joining the points $A(-2,3)$ and $B(3,-2)$ in the ratio $m: n$, then the point which divides $A B$ in the ratio $-4 m: 3 n$ is

32Straight Lines And Pair Of Straight Lines

If the slope of one of the lines represented by $5 x^2+\frac{40}{3} x y+k y^2=0$ is 3 , then the angle between the pair of lines is

33Straight Lines And Pair Of Straight Lines

If $L$ is a line passing through the point $(-1,1)$ and parallel to the common line of the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$, then the equation of pair of lines joining the origin to the points of intersection ...

34Straight Lines And Pair Of Straight Lines

The centroid of the triangle formed by the lines $x-3 y+3=0, x+3 y+3=0 x+y-1=0$ is

35Straight Lines And Pair Of Straight Lines

If a line $L$ is common to the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$ then the combined equation the other two lines is

36Straight Lines And Pair Of Straight Lines

The transformed equation of $3 X^2+4 X Y+Y^2-8 X-4 Y-4=0$ is $f(X, Y)=a X^2+2 h X Y+b Y^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then, $f(1,1)=$

37Straight Lines And Pair Of Straight Lines

Let $O A B C$ be a parallelogram. The equation of one diagonal $A C$ is $x+y-1=0$ and the combined equation of the sides $O A, O C$ is $2 x^2-y^2=0$. If $G$ is centroid of the triangle $O A C$, then $B G=$

38Straight Lines And Pair Of Straight Lines

If the lines drawn along the diagonals of the two squares formed by two pairs of lines $x^2-3|x|+2=0$ and $y^2-3 y+2=0$ form a square $A B C D$, then the equations of two adjacent sides of the square $A B C D$ are

39Straight Lines And Pair Of Straight Lines

The acute angle between the pair of straight lines joining the origin to the points of intersection of the line $x+y-1=0$ with the pair of straight lines $k x^2+8 x y-3 y^2+2 x-4 y-1=0$ is

40Straight Lines And Pair Of Straight Lines

The distance between the point $(2,1)$ and the image of the point $(3,-1)$ with respect to the line $2 x+y-1=0$ is

41Straight Lines And Pair Of Straight Lines

If $\pi / 3$ is the angle between the straight lines $p x+q y+r=0$ and $x \sin \alpha+y \cos \alpha=r(r \neq 0)$ which meet at a point $A$ and the straight line $x \cos \alpha-y \sin \alpha=0$ also passes through the point $A$, then

42Straight Lines And Pair Of Straight Lines

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other, then the points $P(x, y)$ on such lines satisfy

43Straight Lines And Pair Of Straight Lines

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other, then the points $P(x, y)$ on such lines satisfy

44Straight Lines And Pair Of Straight Lines

The area (in square units) of the quadrilateral formed by the point of intersection of the lines $x+y-1=0$, $x-y+1=0$, the point $(1,1)$ and the feet of the perpendiculars from this point on to the lines is

45Straight Lines And Pair Of Straight Lines

For an integer $K$, if the point $P\left(K^2, K+1\right)$ and the origin $O(0,0)$ lie in the same region between the lines $x+2 y-5=0$ and $3 x-y+1=0$, then the possible number of such points $P$ is

46Straight Lines And Pair Of Straight Lines

If $\alpha$ represent the square of the distance between the origin and the point of intersection of the lines $x^2-y^2-x+3 y-2=0$ and $\beta$ represent the product of the perpendicular distances from the origin on the pair of lines, then $...

47Straight Lines And Pair Of Straight Lines

The condition that the lines joining the origin to the points of intersection of the two curves $x^2+y^2+g x+c=0, x^2+y^2+2 f y-c=0$ are at right angles, is

48Straight Lines And Pair Of Straight Lines

If the normal form of the equation of a straight line $4 x+3 y+2=0$ is $x \cos \alpha+y \sin \alpha=p$ and its intercept form is $\frac{x}{a}+\frac{y}{b}=1$, then $\frac{p \sec \alpha}{a b}=$

49Straight Lines And Pair Of Straight Lines

Let $A=(2,3), B=(3,-5)$ be two vertices of $\triangle A B C$ such that $C$ is a point on the line $L \equiv 3 x+4 y-5=0$. Then the locus of the centroid of $\triangle A B C$ is a line parallel to

50Straight Lines And Pair Of Straight Lines

Let $P$ be the point of intersection of the lines $L_1 \equiv x-y-7=0$ and $L_2 \equiv x+y-5=0 . A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ are points on the lines $L_1=0$ and $L_2=0$ respectively such that $P A=3 \sqrt{2}$, $P B=...