Straight Lines — JEE Mains Mathematics

Formulas (3)

• The angle between two lines · Angle between two lines
  $$\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$$
• Intercept and normal forms · Intercept and normal forms
  $$\frac xa+\frac yb=1,\qquad x\cos\alpha+y\sin\alpha=p$$
• Distance along a line · Parametric form
  $$\frac{x-x_1}{\cos\theta}=\frac{y-y_1}{\sin\theta}=r$$

Watch out for (3)

• Two lines, not one
  → The angle between two lines
• Normal form needs a unit normal
  → Intercept and normal forms
• Two directions give two answers
  → Distance along a line

Formulas (3)

• Distance from a point to a line · Perpendicular distance
  $$d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$$
• Parallel lines · Distance between parallel lines
  $$d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$$
• Which side of a line · Same side of a line
  $$(ax_1+by_1+c)(ax_2+by_2+c)>0$$

Watch out for (3)

• Write the line as ax + by + c = 0 first
  → Distance from a point to a line
• Match the coefficients first
  → Parallel lines
• Test with the vertex, not the sketch
  → Which side of a line

Formulas (2)

• Image of a point in a line · Image in a line
  $$\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{2(ax_1+by_1+c)}{a^2+b^2}$$
• Reflected rays and shortest paths · Path via a mirror
  $$AR+RB\ge A'B,\ \text{equality when } R \text{ is on } A'B$$

Watch out for (2)

• Twice, not once
  → Image of a point in a line
• Reflect the right point
  → Reflected rays and shortest paths

Formulas (2)

• Angle bisectors · Angle bisectors
  $$\frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm\frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}}$$
• A pair of lines in one equation · Pair of lines through the origin
  $$ax^2+2hxy+by^2=0:\quad \tan\theta=\frac{2\sqrt{h^2-ab}}{|a+b|}$$

Watch out for (2)

• Origin angle is not always the acute one
  → Angle bisectors
• h is half the xy coefficient
  → A pair of lines in one equation

Formulas (3)

• The orthocentre · Altitude conditions
  $$\overrightarrow{AH}\cdot\overrightarrow{BC}=0,\qquad \overrightarrow{BH}\cdot\overrightarrow{CA}=0$$
• The circumcentre · Equal distances
  $$|OA|^2=|OB|^2=|OC|^2$$
• Centroid, incentre and dividing a side · Incentre
  $$I=\frac{aA+bB+cC}{a+b+c}$$

Watch out for (3)

• Check for a right angle first
  → The orthocentre
• Midpoint alone is not enough
  → The circumcentre
• Weight by the opposite side
  → Centroid, incentre and dividing a side

Formulas (2)

• Area from coordinates · Area of a triangle
  $$\Delta=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$
• Parallelograms, rhombi and isosceles triangles · Parallelogram
  $$A+C=B+D$$

Watch out for (2)

• Two signs from one modulus
  → Area from coordinates
• Which vertices are opposite
  → Parallelograms, rhombi and isosceles triangles

Formulas (2)

• Lines through a fixed point · Family through a point
  $$L_1+\lambda L_2=0$$
• Locus of a moving point · Elimination
  $$h=f(t),\ k=g(t)\ \Rightarrow\ F(h,k)=0$$

Watch out for (2)

• Count every way to fail
  → Lines through a fixed point
• Keep the restrictions
  → Locus of a moving point