Conics — NDA Maths

Formulas (3)

• What a Conic Is — Focus, Directrix, Eccentricity · Focus–directrix definition
  $$\dfrac{PF}{PM} = e$$
• Eccentricity Values & Comparing Conics · Eccentricities
  $$\text{ellipse } e=\sqrt{1-\tfrac{b^2}{a^2}}, \quad \text{hyperbola } e=\sqrt{1+\tfrac{b^2}{a^2}}$$
• Identifying a General Second-Degree Equation · Complete the square to classify
  $$Ax^2 + Cy^2 + Dx + Ey + F = 0 \ \xrightarrow{\text{complete squares}}\ \text{standard form}$$

Watch out for (1)

• A second-degree equation is not always a curve
  → Identifying a General Second-Degree Equation

Formulas (4)

• Standard Forms & Their Elements · Standard parabola
  $$y^2 = 4ax: \ \text{focus } (a,0), \ \text{directrix } x = -a$$
• The Latus Rectum · Length of latus rectum
  $$\text{LR} = 4a, \quad \text{endpoints } (a, \pm 2a)$$
• Focal Distance & Focal Chords · Focal distance
  $$\text{focal distance of } (x_1,y_1) \text{ on } y^2=4ax = x_1 + a$$
• Tangents & Chords of a Parabola · Tangent of slope m
  $$y = mx + \dfrac{a}{m} \quad (\text{to } y^2 = 4ax)$$

Watch out for (1)

• The sign of the linear term sets the direction
  → Standard Forms & Their Elements

Formulas (3)

• Standard Form, Foci & Eccentricity · Foci & eccentricity
  $$c^2 = a^2 - b^2, \qquad e = \dfrac{c}{a}$$
• The Sum of Focal Distances · Constant focal sum
  $$PF_1 + PF_2 = 2a, \qquad \text{latus rectum} = \dfrac{2b^2}{a}$$
• Building the Ellipse from Given Data · Key relations
  $$c = ae, \quad b^2 = a^2 - c^2, \quad \text{LR} = \dfrac{2b^2}{a}$$

Watch out for (1)

• Major axis = larger denominator
  → Standard Form, Foci & Eccentricity

Formulas (2)

• Standard Form, Foci & Eccentricity · Hyperbola foci & eccentricity
  $$c^2 = a^2 + b^2, \qquad e = \dfrac{c}{a} > 1$$
• Parametric Form & θ-Independent Properties · Parametric identity
  $$\sec^2\theta - \tan^2\theta = 1$$

Watch out for (1)

• Hyperbola uses PLUS: c² = a² + b²
  → Standard Form, Foci & Eccentricity