Straight lines practice · Medium

If arithmetic mean of reciprocals of intercepts of line $y=m x+c$ on coordinate axes is $1 / 4$, then this line will pass through the point

If the image of point $\mathrm{P}(2,3)$ in a line $\mathrm{L}$ is $\mathrm{Q}(4,5)$, then the image of point $R(0,0)$ in the same line is:

The distance of the point $(2,3)$ from the line $2 x-3 y$ $+28=0$, measured parallel to the line $\sqrt{3 x}-y+1=0$, is equal to

The line $x + 2y = 4$ is translated parallel to itself by 3 units in the sense of increasing $x$ and then rotated by $30^\circ$ in the clockwise direction about the point where the shifted line cuts the x-axis. The equation of the line in the new position is

The equation to the bisecting the join of $\mathrm{(3, - 4)}$ and (5,2) and having its intercepts on the x-axis and the y-axis in the ratio 2 : 1 is

The equation of a straight line equally inclined to the axes and equidistant from the points $(1,-2)$ and $(3,4)$ is

If $3 a+2 b+6 c=0$, then family of straight lines $a x+b y+c=0$ passes through a fixed point whose coordinates are given by

The equation of bisector of that angle between the lines $x+y+1=0$ and $2 x-3 y-5=0$ which contains the point $(10,-20)$ is

If A and B are two fixed points, then the locus of a point which moves in such a way that the angle APB is a right angle is

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

If $O$ is the origin and $Q$ is a variable point on $x^2=4 y$, then the locus of the mid-point of $O Q$, is

The line which is parallel to x-axis and crosses the curve $y = \sqrt x \;$ at an angle $45^\circ$, is

A line having slope '1' is drawn from a point $A(-3,0)$ cuts a curve $y=x^2+x+1$ at $P \,\&\, Q$. Find $\ell({AP}) \ell({AQ})$.

The line $x-2 y=0$ will be a bisector of the angle between the lines represented by the equation $x^2-2 h x y-2 y^2=0$, if $h=$

If the line $px - qy = r$ intersects the coordinate axes at (a,0) and (0,b), thyen the value of a+b is equal to

A line through the point $A(2,4)$ intersects the line $x+y=9$ at the point $P.$ The minimum distance of $A P,$ is

Family of a lines $x \sec ^2 \theta+y \tan ^2 \theta-2=0$, for different real $\theta$, is

Consider a family of straight lines $(x+y)+\lambda(2 x-y+1)=0$. Equation of the straight line belonging to this family that is farthest from $(1,-3)$ is

A line AB makes zero intercepts on x-axis and y-axis and it is perpendicular to another line CD which is $3x + 4y + 6 = 0$. The equation of line AB is

Number of lines drawn from the point $(4,-5)$ so that its distance from $(-2,3)$ will be equal to 12 are

If the parametric coordinates of a point are $(\tan \theta-\sin \theta, \tan \theta+\sin \theta)$, then the locus of the point is

Let $\theta_{1}$ be the angle between two lines $2 x+3 y+c_{1}=0$ and $-x+5 y+c_{2}=0$ and $\theta_{2}$ be the angle between two lines $2 x+3 y+c_{1}=0$ and $-x+5 y+c_{3}=0$, where $c_{1}, c_{2}, c_{3}$ are any real numbers :
Statement-1: If $c_{2}$ and $c_{3}$ are proportional, then $\theta_{1}=\theta_{2}$. Statement-2: $\theta_{1}=\theta_{2}$ for all $c_{2}$ and $c_{3}$.

If the points $(-2,0),\left(-1, \frac{1}{\sqrt{3}}\right)$ and $(\cos \theta, \sin \theta)$ are collinear, then the number of values of $\theta \in[0,2 \pi]$ is

The parallelism condition for two straight lines one of which is specified by the equation $ax + by + c = 0$ and the other being represented parametrically by $x = \mathrm{\alpha }t + \mathrm{\beta },\mathrm{\;}y = \mathrm{\gamma }t + \mathrm{\delta },$ is given by

The measure of the angle between the lines $x \cos \beta+y \sin \beta=\frac{5 \pi}{2}$ and $x \cos \alpha+y \sin \alpha=\frac{\pi}{2}$ is

If $P=(1,0), Q=(-1,0)$ and $R=(2,0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^2+S R^2=2 S P^2$, is

The value of $\lambda$ for which the lines joining the point of intersection of curves $C_1$ and $C_2$ to the origin are equally inclined to the $X$-axis, where $C_1: \lambda x^2+3 y^2-2 \lambda x y+9 x=0$ and $C_2: 3 x^2-4 y^2+8 x y-3 x=0$, is

The image of the point (4,-3) with respect to the line y=x is

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y+8=0$, then the value of $g+c+h-f$ equals

The acute angle formed between the lines joining the origin to the points of intersection of the curves $x^2+y^2-2 x-1=0$ and $x+y=1$, is