Slope of a Line

The vertices of a triangle $∆OBC$ are $O \left(0, 0\right), B\left(-3, -1\right), C\left(-1, -3\right)$ find the equation of the line parallel to $BC$ and intersecting the sides $OB \& OC$, whose perpendicular distance from the point $\left(0, 0\right)$ is half.

Points $P\left(-3,2\right), Q\left(9,10\right) \text{and} R\left(\alpha ,4\right)$ lie on a circle $C$ with $PR$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S.$ If $S$ lies on the line $2x-ky=1,$ then $k$ is equal to $?$

If $Q\left(h, k\right)$ is the foot of the perpendicular from $\mathrm{P}\left({\mathrm{x}}_1, {\mathrm{y}}_1\right)$ on the straight line $ax+by+c=0$ then $\left(h-x_1\right):a=\left(k-y_1\right):b=-\left(a{x}_1+b{y}_1+c\right):\left(a^2+b^2\right)$

Find the slope of the line making inclination of $60°$ with the positive direction of $x$-axis.

Find the slope of the line passing through the points $\left(3, -2\right)$ and $\left(3, 4\right)$.

Find the slope of the line passing through the points $\left(3, -2\right)$ and $\left(7, -2\right)$,

Find the slope of the line passing through the points $\left(3,-2\right)$ and $\left(-1, 4\right)$.

If $\mathrm{A}\left(6,1\right), \mathrm{B}\left(8,2\right), \mathrm{C}\left(9,4\right)$ and $\mathrm{D}\left(7,3\right)$ are the vertices of $\square \mathrm{ABCD}$. Using slopes, show that $\square \mathrm{ABCD}$ is a parallelogram.

The acute angles between the lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ is $\theta ,$ then find $\cos \theta .$

Find the abscissa of points of contact of $y=x^3-3x$ whose tangents passes through $\left(2,-2\right)$.

Line segment $AB$ of fixed length $c$ slides between coordinate axes such that its ends $A$ and $B$ lie on the axes. If $O$ is origin and and rectangle $OAPB$ is completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $AB$ is $x^{\frac{2}{3}}+y^{\frac{2}{3}}=c^{\frac{2}{3}}.$

If one side of a square in parallel to $12x+5y=0$ & its area being $36$ while centre being $\left(1,-1\right)$, then find equation of sides of parallelogram.

Find the angle between the lines $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6=0$.

In Figure, time and distance graph of a linear motion is given. Two positions of time and distance are recorded as, when $T=0, D=2$ and when $T=3, D=8$. Using the concept of slope, find law of motion, i.e., how distance depends upon time.

Three points $P\left(h, k\right), Q\left(x_1, y_1\right)$ and $R\left(x_2, y_2\right)$ lie on a line. Show that $\left(h-x_1\right)\left(y_2-y_1\right)=\left(k-y_1\right)\left(x_2-x_1\right)$

Line through the points $\left(-2, 6\right)$ and $\left(4, 8\right)$ is perpendicular to the line through the points $\left(8, 12\right)$ and $\left(x, 24\right)$. Find the value of $x$.

If the angle between two lines is $\frac{\pi }{4}$ and slope of one of the lines is $\frac{1}{2}$, find the slope of the other line.

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

Show that the slope of a line which is parallel to $y$-axis is not defined.

Find the slope of a line which is parallel to $x$-axis.