Parabola

a. Standard equation of a parabola:

• General equation of a parabola: (y^2=4ax)
• Vertex: ((0, 0))
• Focus: ((a, 0))
• Directrix: (x = -a)

b. Properties of a parabola:

• Focus is equidistant from the vertex and any point on the parabola.
• The tangent at any point on the parabola makes equal angles with the axis of symmetry and the line joining the point to the focus.
• The subnormal at any point on the parabola is constant and equal to the latus rectum.

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

c. Equations of tangents and normals to a parabola:

• Equation of a tangent to the parabola (y^2=4ax) at the point ((x_1, y_1)): $$y – y_1 = m(x – x_1),$$ where (m) is the slope of the tangent.
• Equation of a normal to the parabola (y^2=4ax) at the point ((x_1, y_1)): $$y – y_1 = - \frac{1}{m}(x – x_1),$$ where (m) is the slope of the tangent.

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

d. Equations of common tangents to two parabolas:

• Consider two parabolas (y^2=4a_1x) and (y^2=4a_2x).
• Equation of the common tangent to the two parabolas: $$yy_2(a_1-a_2) = x(y_1^2-y_2^2)$$

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

Ellipse

a. Standard equation of an ellipse:

• General equation of an ellipse: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, (a>b)$$
• Centre: ((0, 0))
• Vertices: ((\pm a, 0) )
• Co-vertices: ((0, \pm b ))
• Foci: ((\pm c, 0)), where (c^2=a^2-b^2)

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

b. Properties of an ellipse:

• The sum of the distances of any point on the ellipse from the two foci is constant and equal to (2a).
• The tangent at any point on the ellipse makes equal angles with the focal radii to that point.
• The product of the lengths of the semi-major and semi-minor axes is constant and equal to (a^2-b^2).

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

c. Equations of tangents and normals to an ellipse:

• Equation of a tangent to the ellipse (\frac{x^2}{a^2}+\frac{y^2}{b^2}=1) at the point ((x_1, y_1)): $$ \frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1$$

• Equation of a normal to the ellipse (\frac{x^2}{a^2}+\frac{y^2}{b^2}=1) at the point ((x_1, y_1)): $$ \frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1$$ References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

d. Equations of conjugate diameters:

• If the equation of a chord of the ellipse (\frac{x^2}{a^2}+\frac{y^2}{b^2}=1) is $$lx+my=1$$ then the equation of the conjugate diameter is $$lx-my=1$$ References:

• NCERT Mathematics Class 12, Chapter 6: Conic Sections

Hyperbola

a. Standard equation of a hyperbola:

• General equation of a hyperbola: $$ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
• Center: ((0,0))
• Vertices: ((\pm a, 0))
• Foci: ((\pm c, 0)), where (c^2 = a^2+b^2)
• Transverse axis: (2a)
• Conjugate axis: (2b)

References:

• NCERT Mathematics Class 11, Chapter 10: Conic Sections

b. Properties of a hyperbola:

• The difference between the distances of any point on the hyperbola from the two foci is constant and equal to (2a).
• The tangent at any point on the hyperbola makes equal angles with the focal radii to that point.
• The product of the lengths of the semi-transverse and semi-conjugate axes is constant and equal to (a^2 - b^2).
• Asymptotes of hyperbola: ( y = \pm \frac{b}{a}x )

References:

• NCERT Mathematics Class 11, Chapter 1