Notes from Toppers
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: $$yy_2(a_1a_2) = x(y_1^2y_2^2)$$
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) )
 Covertices: ((0, \pm b ))
 Foci: ((\pm c, 0)), where (c^2=a^2b^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 semimajor and semiminor axes is constant and equal to (a^2b^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 $$lxmy=1$$ 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 semitransverse and semiconjugate 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