Straight Line — MHT-CET Maths

Formulas (5)

• Slope and Inclination: Parallel Means m₁ = m₂, Perpendicular Means m₁m₂ = −1 · Slope
  $$m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1} = -\frac{a}{b} \qquad \text{parallel: } m_1 = m_2 \qquad \text{perpendicular: } m_1 m_2 = -1$$
• Angle Between Two Lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)| · Angle between lines
  $$\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \qquad \text{one line vertical: } \tan\theta = \left|\frac{1}{m}\right|$$
• Lines Through a Point at a Given Angle: Solve the Angle Formula for m · Two slopes
  $$\left|\frac{m - m_1}{1 + m m_1}\right| = \tan\alpha \ \Rightarrow\ m = \frac{m_1 \pm \tan\alpha}{1 \mp m_1\tan\alpha} \quad(\text{inclinations } \theta_1 \pm \alpha)$$
• Rotating a Line About a Point: Add or Subtract the Angle From the Inclination · Rotation
  $$\theta_{\text{new}} = \theta \pm \alpha \ (+\text{ anticlockwise}),\qquad y - y_0 = \tan\theta_{\text{new}}\,(x - x_0)$$
• Bisector of an Angle at a Vertex, and the Reflection of a Line · Bisector direction
  $$\theta_{\text{bisector}} = \frac{\theta_{QP} + \theta_{QR}}{2} \qquad \text{reflected slope: the other root of } \left|\frac{m - m_L}{1 + m m_L}\right| = \tan\theta$$

Watch out for (5)

• Reading a negative slope as a negative angle
  → Slope and Inclination: Parallel Means m₁ = m₂, Perpendicular Means m₁m₂ = −1
• Dropping the modulus and reporting the obtuse angle
  → Angle Between Two Lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
• Keeping only one root
  → Lines Through a Point at a Given Angle: Solve the Angle Formula for m
• Rotating the wrong way
  → Rotating a Line About a Point: Add or Subtract the Angle From the Inclination
• Bisecting the inclinations instead of the directions
  → Bisector of an Angle at a Vertex, and the Reflection of a Line

Formulas (5)

• Intercept Form x/a + y/b = 1: Ratios of Intercepts, Triangle Area and 1/a² + 1/b² = 1/p² · Intercept form
  $$\frac{x}{a} + \frac{y}{b} = 1,\qquad \text{area with axes} = \frac{|ab|}{2},\qquad \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{p^2}$$
• Normal Form x cos α + y sin α = p: the Perpendicular's Length and Direction · Normal form
  $$x\cos\alpha + y\sin\alpha = p \quad (p > 0),\qquad \text{foot of the perpendicular } (p\cos\alpha, p\sin\alpha)$$
• Point-Slope and Two-Point Forms: Medians, Parallels Through a Point, and Reading Intercepts Back · Point-slope
  $$y - y_1 = m(x - x_1),\qquad \text{midpoint } \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
• Intersection of Two Lines, and a Line Through It With a Given Property · Family through an intersection
  $$L_1 + \lambda L_2 = 0 \qquad \text{equal intercepts: } x + y = c$$
• Concurrency of Three Lines: the 3 × 3 Determinant Is Zero · Concurrency
  $$\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0$$

Watch out for (5)

• Forgetting the negative-product case
  → Intercept Form x/a + y/b = 1: Ratios of Intercepts, Triangle Area and 1/a² + 1/b² = 1/p²
• Matching the distance and ignoring the direction
  → Normal Form x cos α + y sin α = p: the Perpendicular's Length and Direction
• Using the vertex instead of the midpoint
  → Point-Slope and Two-Point Forms: Medians, Parallels Through a Point, and Reading Intercepts Back
• Counting a divisor that gives a fractional m
  → Intersection of Two Lines, and a Line Through It With a Given Property
• Expanding the determinant wrong and losing the k term
  → Concurrency of Three Lines: the 3 × 3 Determinant Is Zero

Formulas (2)

• Section Formula: Internal and External Division · Section formula
  $$P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right) \quad(\text{internal});\qquad \frac{AO}{OB} = \frac{d_1}{d_2} \text{ between parallel lines}$$
• Midpoints Everywhere: Rectangle Centres, Missing Vertices and the Circumcentre · Rectangle centre
  $$\text{centre} = \text{midpoint of a diagonal};\qquad \text{right angle at a vertex: } m_{VA}\, m_{VC} = -1$$

Watch out for (2)

• Swapping the weights
  → Section Formula: Internal and External Division
• Solving for the vertices without fixing c first
  → Midpoints Everywhere: Rectangle Centres, Missing Vertices and the Circumcentre

Formulas (4)

• Distance From a Point to a Line · Point to line
  $$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$
• Distance Between Parallel Lines, and the Square Between Them · Parallel lines
  $$d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \quad\text{for } ax + by + c_1 = 0,\ ax + by + c_2 = 0$$
• Foot of the Perpendicular: Intersect the Perpendicular Through the Point With the Line · Foot of the perpendicular
  $$\frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2}$$
• Distance From a Point to a Line Measured Parallel to Another Line · Along a direction
  $$d_\theta = \frac{|ax_1 + by_1 + c|}{|a\cos\theta + b\sin\theta|}$$

Watch out for (4)

• Comparing |c| without dividing by √(a² + b²)
  → Distance From a Point to a Line
• Subtracting constants of differently scaled lines
  → Distance Between Parallel Lines, and the Square Between Them
• Sign slips in the ratio form
  → Foot of the Perpendicular: Intersect the Perpendicular Through the Point With the Line
• Reporting the perpendicular distance
  → Distance From a Point to a Line Measured Parallel to Another Line