Straight Lines — NDA Mathematics

Formulas (3)

• Slope and the forms of a line · Slope and forms of a line
  $$m=\dfrac{y_2-y_1}{x_2-x_1}\qquad m_{ax+by+c=0}=-\dfrac{a}{b}\qquad y-y_1=m(x-x_1)\qquad y=mx+c$$
• Intercept form and intercepts · Intercept form and intercepts
  $$\dfrac{x}{a}+\dfrac{y}{b}=1\qquad x\text{-intercept}=-\dfrac{c}{a}\qquad y\text{-intercept}=-\dfrac{c}{b}$$
• Family of lines and concurrency · Family of lines and concurrency
  $$L_1+\lambda L_2=0\qquad \begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}=0$$

Watch out for (4)

• Slope is $\Delta y/\Delta x$, not $\Delta x/\Delta y$ — and a vertical line has undefined slope
  → Slope and the forms of a line
• Intercept form needs the constant on the **RHS as $1$** — $a,b$ are the intercepts only then
  → Intercept form and intercepts
• The pencil is $L_1+\lambda L_2=0$ — keep each $L_i$ in the form $=0$ first
  → Family of lines and concurrency
• The foot of the perpendicular is the *midpoint* of $PP'$, not the image itself
  → Image of a point and reflections

Formulas (2)

• Distance: point-point, point-line, parallel lines · Distance formulas
  $$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\qquad \dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}\qquad \dfrac{|c_1-c_2|}{\sqrt{a^2+b^2}}$$
• Section formula and ratios · Section formula and midpoint
  $$\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)\qquad \left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$$

Watch out for (3)

• Put the line in $ax+by+c=0$ form first, and divide by $\sqrt{a^2+b^2}$ — not by $a+b$
  → Distance: point-point, point-line, parallel lines
• In $m:n$ the weight $m$ multiplies the far endpoint $x_2$ — mind the cross-pairing
  → Section formula and ratios
• Equidistant from two points gives a line (perp. bisector); equidistant from two lines gives the angle bisectors
  → Locus from a condition

Formulas (2)

• Angle between two lines · Angle between two lines
  $$\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right|$$
• Parallel and perpendicular conditions · Parallel and perpendicular conditions
  $$\text{Parallel: } m_1=m_2\qquad \text{Perpendicular: } m_1 m_2=-1\qquad a_1a_2+b_1b_2=0$$

Watch out for (2)

• The difference of slopes is on top: $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$
  → Angle between two lines
• Perpendicular slope is the negative reciprocal: $m_2=-\dfrac{1}{m_1}$, not $\dfrac{1}{m_1}$
  → Parallel and perpendicular conditions

Formulas (4)

• Area of a triangle and collinearity · Area of a triangle from vertices
  $$\text{Area}=\dfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$
• Centroid, incentre, circumcentre · Centroid and incentre
  $$G=\left(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right)\qquad I=\dfrac{a\,A+b\,B+c\,C}{a+b+c}$$
• Constructing a triangle: vertices, medians, altitudes · Vertex from a midpoint
  $$B+C=2M\qquad C=2M-B$$
• Parallelograms, squares and diagonals · Parallelogram: fourth vertex and area
  $$A+C=B+D\qquad D=A+C-B\qquad \text{Area}=|x_1y_2-x_2y_1|$$

Watch out for (4)

• Don't forget the $\tfrac12$ and the absolute value — and collinearity is area $=0$
  → Area of a triangle and collinearity
• Centroid is the plain average; the incentre weights by the opposite side lengths $a,b,c$
  → Centroid, incentre, circumcentre
• An altitude is perpendicular to the opposite side — use the negative-reciprocal slope
  → Constructing a triangle: vertices, medians, altitudes
• In parallelogram $ABCD$ the diagonals are $AC$ and $BD$: $A+C=B+D$
  → Parallelograms, squares and diagonals