\displaystyle \textbf{1. Equation of a Straight Line}
\displaystyle \text{Every first degree equation in }x,y\text{ represents a straight line.}
\\

\displaystyle \textbf{2. Slope of a Line}
\displaystyle \text{The trigonometric tangent of the angle that a non-vertical line makes with the positive direction}
\displaystyle \text{of the x-axis in anticlockwise sense is called the slope or gradient of the line.}
\\

\displaystyle \textbf{3. Formula for Slope}
\displaystyle \text{The slope }m\text{ of a non-vertical line passing through }(x_1,y_1)\text{ and }(x_2,y_2)\text{ is given by}
\displaystyle m=\frac{y_2-y_1}{x_2-x_1}
\displaystyle =\frac{\text{Difference of ordinates}}{\text{Difference of abscissae}}
\\

\displaystyle \textbf{4. Slope of Horizontal and Vertical Lines}
\displaystyle \text{The slope of a horizontal line is }0\text{ and the slope of a vertical line is undefined.}
\\

\displaystyle \textbf{5. Angle Between Two Lines}
\displaystyle \text{If two lines have slopes }m_1\text{ and }m_2,\text{ then the acute angle }\theta\text{ between them is}
\displaystyle \tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|,\;1+m_1m_2\neq0
\\

\displaystyle \textbf{6. Parallel Lines}
\displaystyle \text{Two lines are parallel iff their slopes are equal, i.e., }m_1=m_2.
\\

\displaystyle \textbf{7. Perpendicular Lines}
\displaystyle \text{Two lines are perpendicular iff the product of their slopes is }-1.
\displaystyle m_1m_2=-1
\\

\displaystyle \textbf{8. Collinearity of Three Points}
\displaystyle \text{Three points }P,Q,R\text{ are collinear iff}
\displaystyle \text{Slope of }PQ=\text{Slope of }QR.
\\

\displaystyle \textbf{9. Intercepts on Axes}
\displaystyle \text{If a straight line cuts the x-axis at }A\text{ and the y-axis at }B,\text{ then }OA\text{ and }OB
\displaystyle \text{are called the intercepts of the line on x-axis and y-axis respectively.}
\\

\displaystyle \textbf{10. Line Parallel to the x-axis}
\displaystyle \text{The equation of a line parallel to x-axis at a distance }a\text{ from it is}
\displaystyle y=a\quad\text{or}\quad y=-a.
\\

\displaystyle \textbf{11. Line Parallel to the y-axis}
\displaystyle \text{The equation of a line parallel to y-axis at a distance }b\text{ from it is}
\displaystyle x=b\quad\text{or}\quad x=-b.
\\

\displaystyle \textbf{12. Equation of the x-axis}
\displaystyle y=0
\\

\displaystyle \textbf{13. Equation of the y-axis}
\displaystyle x=0
\\

\displaystyle \textbf{14. Slope-Intercept Form}
\displaystyle \text{The equation of a line with slope }m\text{ and y-intercept }c\text{ is}
\displaystyle y=mx+c.
\\

\displaystyle \textbf{15. Line Through the Origin}
\displaystyle \text{The equation of a line with slope }m\text{ passing through the origin is}
\displaystyle y=mx.
\\

\displaystyle \textbf{16. Point-Slope Form}
\displaystyle \text{The equation of the line passing through }(x_1,y_1)\text{ and having slope }m\text{ is}
\displaystyle y-y_1=m(x-x_1).
\\

\displaystyle \textbf{17. Two-Point Form}
\displaystyle \text{The equation of the line passing through }(x_1,y_1)\text{ and }(x_2,y_2)\text{ is}
\displaystyle y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1).
\\

\displaystyle \textbf{18. Intercept Form}
\displaystyle \text{The equation of the line making intercepts }a\text{ and }b\text{ on the x-axis and y-axis respectively is}
\displaystyle \frac{x}{a}+\frac{y}{b}=1.
\\

\displaystyle \textbf{19. Normal Form}
\displaystyle \text{The equation of the straight line upon which the length of the perpendicular from the origin is }
\displaystyle p\text{ and the angle between this perpendicular and the positive x-axis is }\alpha\text{ is}
\displaystyle x\cos\alpha+y\sin\alpha=p.
\\

\displaystyle \textbf{20. Line Through a Point Making an Angle } \theta
\displaystyle \text{The equation of the straight line passing through }(x_1,y_1)\text{ and making an angle }\theta
\displaystyle \text{with the positive direction of x-axis is}
\displaystyle \frac{x-x_1}{\cos\theta}=\frac{y-y_1}{\sin\theta}=r
\displaystyle \text{where }r\text{ is the distance of the point }(x,y)\text{ on the line from }(x_1,y_1).
\displaystyle \text{The coordinates of any point on the line at a distance }r\text{ from }(x_1,y_1)\text{ are}
\displaystyle (x_1\pm r\cos\theta,\;y_1\pm r\sin\theta).
\\

\displaystyle \textbf{21. Slope of } ax+by+c=0
\displaystyle \text{The slope of the line }ax+by+c=0\text{ is}
\displaystyle -\frac{a}{b}.
\\

\displaystyle \textbf{22. Concurrent Lines}
\displaystyle \text{Three lines }
\displaystyle L_1=a_1x+b_1y+c_1=0,\;L_2=a_2x+b_2y+c_2=0,\;L_3=a_3x+b_3y+c_3=0
\displaystyle \text{are concurrent if}
\displaystyle \left|\begin{array}{ccc}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right|=0.
\displaystyle \text{Also, they are concurrent if there exist scalars }\lambda_1,\lambda_2,\lambda_3\text{ such that}
\displaystyle \lambda_1L_1+\lambda_2L_2+\lambda_3L_3=0.
\\

\displaystyle \textbf{23. Line Parallel to } ax+by+c=0
\displaystyle \text{The equation of a line parallel to }ax+by+c=0\text{ is}
\displaystyle ax+by+\lambda=0,
\displaystyle \text{where }\lambda\text{ is a constant.}
\\

\displaystyle \textbf{24. Line Perpendicular to } ax+by+c=0
\displaystyle \text{The equation of a line perpendicular to }ax+by+c=0\text{ is}
\displaystyle bx-ay+\lambda=0,
\displaystyle \text{where }\lambda\text{ is a constant.}
\\

\displaystyle \textbf{25. Perpendicular Distance of a Point from a Line}
\displaystyle \text{The perpendicular distance }d\text{ of the line }ax+by+c=0\text{ from the point }(x_1,y_1)\text{ is}
\displaystyle d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}.
\\

\displaystyle \textbf{26. Distance Between Two Parallel Lines}
\displaystyle \text{The distance between the parallel lines }ax+by+c_1=0\text{ and }ax+by+c_2=0\text{ is}
\displaystyle d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}.
\\

\displaystyle \textbf{27. Lines Making an Angle } \alpha \text{ with } y=mx+c
\displaystyle \text{The equations of the lines passing through }(x_1,y_1)\text{ and making an angle }\alpha
\displaystyle \text{with the line }y=mx+c\text{ are given by}
\displaystyle y-y_1=\frac{m\pm\tan\alpha}{1\mp m\tan\alpha}(x-x_1).
\\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.