The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3},$ find the slopes of the lines.

1. Distance Formula:

(i) The distance between two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ is given by $PQ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

(ii) The distance of a point $P\left(x,y\right)$ from the origin $O\left(0,0\right)$ is given by $OP=\sqrt{x^2+{\mathrm{y}}^2}$

2. Area of a triangle given three vertices:

(i) The area of the triangle, the coordinates of whose vertices are $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ is$=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$ or $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& \mathrm{ }{y}_3& 1\end{array}\right|$

(ii) If the points (x1,y1), $(x_2,y_2)$ and $(x_3,y_3)$ are collinear, then $\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& \mathrm{ }{y}_3& 1\end{array}\right|=0$

3. Section Formula:

The coordinates of a point dividing a line joining the points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ in the ratio $m:n$ is given by

(i) For internal division:

$\left(\frac{\mathrm{ \; \; m}{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$

(ii) For external division:

     $\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n}\right)$

(iii) Mid-point of a line segment: The coordinates of the mid- point of the line segment joining (x1,y1) and $(x_2,y_2)$ are x1+x22,y1+y22.

4. Co-ordinates of some particular points:

Let $A\left(x_1,y_1\right),B\left(x_2,y_2\right),$ and $C\left(x_3,y_3\right)$ be the vertices of any triangle $ABC$, then

(i) Co-ordinates of Centroid$=G\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

(ii) Co-ordinates of Incentre$=I\left(\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c},\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c}\right)$

(iii) Co-ordinates of Circumcentre$$$=O\left(\frac{x_1\sin 2A+x_2\sin 2B+x_3\sin 2C}{\sin 2A+\sin 2B+\sin 2C},\frac{y_1\sin 2A+y_2\sin 2B+y_3\sin 2C}{\sin 2A+\sin 2B+\sin 2C}\right)$

$$5. Definition of a Straight line: Every first-degree equation in $x, y$ represents a straight line.

6. Slope of a line:

(i) The slope of a line usually denoted by $m$, with the angle of inclination $\theta$ is defined as $\tan \theta$. The slope of a line is also termed as the gradient of a line.

(ii) The slope $\tan \theta$ of a non−vertical line passing through the points $P\left(x_1,y_1\right)$ and $Q\left(x_1,y_1\right)$ is given by $\tan \theta =\frac{y_2-y_1}{x_2-x_1}$.

(iii) Slope of a horizontal line is zero and the slope of a vertical line is not defined.

7. Angle between two lines:

An acute angle $\theta$ between the lines having slopes $m_1=\tan {\alpha }_1$ and $m_2=\tan {\alpha }_2$ is given by $\tan \theta =\frac{m_2-m_1}{1+m_1{m}_2}$.

8. Condition for Parallel and Perpendicular Lines:

(i) Two lines are parallel if and only if their slopes are equal.

(ii) Two lines are perpendicular if and only if the product of their slopes is $-1$.

9. Collinearity of three points:

Three points $A,B$ and $C$ are collinear if and only if Slope of $AB=$Slope of $BC$.

10. Various Forms of the Equation of a Line:

(i) Horizontal and vertical lines:

(a) The equation of a line parallel to $x$−axis at a distance $a$ from it is $y=a$ or $y=-a$ according as it is above or below $x$−axis.

(b) The equation of a line parallel to $y$−axis at a distance $b$ from it is $x=b$$$or $x=-b$ according as it is on the right or on the left side of $y$−axis.

(c) The equation of $y$−axis is $x=0$.

(d) The equation of $x$−axis is $y=0$.

(ii) Slope-intercept form:

The equation of a line with slope $m$ and making an intercept $c$ on $y$−axis is $y=mx+c$.

(iii) Point-slope form:

The equation of the line which passes through the point $P\left(x_1,y_1\right)$ and has slope $m$ is $y-y_1=m\left(x-x_1\right)$.

(iv) Two-point form:

The equation of the line passing through the points $P_1\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ is $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$.

(v) Intercept-form:

The equation of the line making intercepts $a$ and $b$ on $x$-axis and $y$−axis respectively is $\frac{x}{a}+\frac{y}{b}=1$.

(vi) Normal-form:

The equation of the straight line upon which the length of the perpendicular from the origin is $p$ and the angle between this perpendicular and positive $x$ −axis is $\alpha$ is given by $x\cos \alpha +y\sin \alpha =p$.

11. The slope of the line$ax+by+c=0$ is $-\frac{a}{b}$.

12. Concurrency of three lines:

Three lines $y=m_{1}x+c_1, y=m_{2}x+c_2$ and $y=m_{3}x+c_3$ are concurrent, if $m_1\left(c_2-c_3\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right)=0$.

13. Line Parallel to a Given Line:

The equation of a line parallel to the line $ax+by+c=0$ is $ax+by+\lambda =0$, where $\lambda$ is a constant.

14. Line Perpendicular to a Given Line:

The equation of a line perpendicular to the line $ax+by+c=0$ is $bx-ay+\lambda =0$, where $\lambda$ is a constant.

15. Distance of a Point From a Line:

The perpendicular distance $\left(d\right)$ of a line $ax+by+c=0$ from a point $\left(x_1,y_1\right)$ is given by $d=\frac{\left|a{x}_1+b{y}_1+c\right|}{\sqrt{a^2+b^2}}$.

16. Distance Between Two Parallel Lines:

The distance $\left(d\right)$ between the lines $ax+by+c_1=0$ and $ax+by+c_2=0$ is given by $d=\frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}}$

17. Equation of Family of Lines Passing through the Point of Intersection of Two Lines:

The equation $L_1+\lambda L_2=0$(where $\lambda$ is a parameter) is the equation of the family of lines passing through the intersection of $L_1=0$ and $L_2=0$.