Straight lines practice · Mixed difficulty

If the line $p{x^2} - qxy - {y^2} = 0$ makes an angles $\alpha$ and $\beta$ with x-axis, then the value of tan $\left( {\alpha + \beta } \right)$ is

Equation of the line passing through the point ( $a-1$, $a+1)$ and making zero intercept on both axes is

Let ${ax}+{by}+{c}=0$ be a variable straight line, where ${a}, {b}$ and ${c}$ are $1^{\text {st}}, 3^{\text {rd}}$ and $7^{\text {th}}$ terms of an increasing A.P. respectively. Then the variable straight line always passes through a fixed point which lies on-

The algebraic sum of the perpendicular distances from $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$ to a variable line is zero, then the line passes through

The locus of the point of intersection of lines $x\cos \alpha + y\sin \alpha = a$ and $x\sin \alpha - y\cos \alpha = b$ is ($\alpha$ is a variable)

If $t_1, t_2$ and $t_3$ are distinct, the points $\left(t_1, 2 a t_1+a t_1^3\right)$, $\left(t_2, 2 a t_2+a t_2^3\right)$ and $\left(t_3, 2 a t_3+a t_3^3\right)$ are collinear if

A line through the point A(2,0) which makes an angle of ${30^ \circ }$ with the positive direction of x-axis is rotated about A in clockwise direction through an angle of ${15^ \circ }$. Then, the equation of the straight line in the new position is

If non-zero numbers a,b,c are in HP, then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0\;$ always passes through a fixed point. That point is

The image of the point (3,8) in the line $x + 3y = 7,$ is

The equations of sides of a triangle $A B C$ are $A B: x+y-1=0, B C: 7 x-y-15=0, A C: x-y-1=0$ the equation of angular bisector of internal angle $B$ of the triangle is $3 x$ $+y-k=0$. The positive integer $k$ must be.

The image of the point $(-1,3,4)$ in the plane $x-2 y=0$ is

Let a variable line passing through the centre of the circle $x^2+y^2-16 x-4 y=0$, meet the positive coordinate axes at the points $A$ and $B$. Then the minimum value of $O A+O B$, where $O$ is the origin, is equal to

The lines ${(lx + my)^2} - 3{(mx - ly)^2} = 0$ and $lx + my + n = 0$ form

The equation of the line passing through the origin and the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} + \frac{y}{a} = 1$ is

The set of lines $a x+b y+c=0$, where $3 a+2 b+4 c=0$ is concurrent at the point

(i) Find the integral values of '$h$' for which $h x^2-5 x y+4 {hy}^2+{x}+2 {y}-2=0$ represents two real straight lines.
(ii) If the pair of lines represented by equation $k(k-3) x^2+16 x y+(k+1) y^2=0$ are perpendicular to each other, then find $k$.

If the image of a point $P(2,3)$ in the line mirror $y=x$ is the point $Q$ and the image $Q$ in the line mirror $y=0$ is the point $R(x, y)$, then the coordinates of $R$ are

The equation of a line passing through $\left( { - 2,\; - 4} \right)$ and perpendicular to the line $3x - y + 5 = 0$ is

If the angle between the lines $\sqrt{3 y}-x+4=0$ and $x+y-6=0$ is $\theta$, then $\tan \theta$ is equal to.

The straight lines $x + y = 0,\;3x + y - 4 = 0$ and $x + 3y - 4 = 0$ form a triangle which is

The image of the point (1,3) in the line $x + y - 6 = 0,$ is

One possible condition for the three points $\left( {a,\;b} \right),\;\left( {b,\;a} \right)$ and $\left( {{a^2},\; - {b^2}} \right)$ to be collinear, is

The nearest point on the line $3x - 4y = 25$ from the origin is

The diagonals of the parallelogram whose sides are $lx + my + n = 0,\;lx + my + n' = 0,\;mx + ly + n = 0,\;mx + ly + n' = 0$ include an angle

A triangle with vertices $\left( {4,\;0} \right),\;\left( { - 1,\; - 1} \right),\;\left( {3,\;5} \right)$ is

The equations $ax + by + c = 0$ and $dx + ey + f = 0$ represent the same straight line if and only if

If the point $P\left(a, a^2\right)$ lies inside the triangle formed by the lines $x=0, y=0$ and $x+y=2$, then exhaustive range of '$a$' is

The value of $k$ for which the lines $2x - 3y + k = 0,3x - 4y - 13 = 0$ and $8x - 11y - 33 = 0$ are concurrent, is

The distance of the $x + y - 8 = 0$ and $\left( {4,1} \right)$ measured along the direction whose slope is -2 is

$y=x+r$ and $y=-x+r$ where $r$ takes all decimal digits. Then the number of squares in $x y$ plane formed by these lines with diagonals of 2 units length are