Parabola Formula Sheet

A parabola is a U-shaped curve that is the graph of a quadratic function. It is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Neetesh Kumar | June 25, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Conic Sections
• 2. General Equation of a Conic
• 3. Distinguishing Between the Conic
• 4. Parabola
• 5. Parametric Representation
• 6. Type of Parabola
• 7. Position of a Point Relative to a Parabola
• 8. Chord Joining Two Points
• 9. Line & A Parabola
• 10. Length of Subtangent & Subnormal
• 11. Tangent to the Parabola
• 12. Normal to the Parabola
• 13. Pair of Tangents
• 14. Chord of Contact
• 15. Chord with a Given Middle Point
• 16. Diameter
• 17. Co-normal Points
• 18. Important Highlights

1. Conic Sections

A conic section, or conic, is the locus of a point that moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.

• The fixed point is called the FOCUS.
• The fixed straight line is called the DIRECTRIX.
• The constant ratio is called the ECCENTRICITY denoted by $e$.
• The line passing through the focus & perpendicular to the directrix is called the AXIS.
• A point of intersection of a conic with its axis is called a VERTEX.

2. General Equation of a Conic: Focal Directrix Property

The general equation of a conic with focus $(p, q)$ & directrix $lx + my + n = 0$ is:
$(l^2 + m^2) [(x – p)^2 + (y – q)^2] = e^2 (lx + my + n)^2$ $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$

3. Distinguishing Between the Conic

The nature of the conic section depends upon the position of the focus $S$ w.r.t. the directrix & also upon the value of the eccentricity $e$. Two different cases arise.

Case (i) When the focus lies on the directrix:

In this case, $D = abc + 2fgh – af^2 – bg^2 – ch^2 = 0$ & the general equation of a conic represents a pair of straight lines and if:

• $e > 1$, $h^2 > ab$ the lines will be real & distinct intersecting at $S$.
• $e = 1$, $h^2 = ab$ the lines will coincident.
• $e < 1$, $h^2 < ab$ the lines will be imaginary.

Case (ii) When the focus does not lie on the directrix:

The conic represents:

• a parabola: $e = 1; D \neq 0$ or $h^2 = ab$
• an ellipse: $0 < e < 1; D \neq 0$ or $h^2 < ab$
• a hyperbola: $e > 1; D \neq 0$ or $h^2 > ab$
• a rectangular hyperbola: $e > 1; D \neq 0; a + b = 0$ or $h^2 > ab; a+b = 0$

4. Parabola

A parabola is the locus of a point that moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix). The standard equation of a parabola is $y^2 = 4ax$. For this parabola:

• Vertex is $(0, 0)$
• Focus is $(a, 0)$
• Axis is $y = 0$
• Directrix is $x + a = 0$

Properties:

• Focal Distance: The distance of a point on the parabola from the focus.
• Focal Chord: A chord of the parabola which passes through the focus.
• Double Ordinate: A chord of the parabola perpendicular to the axis of the symmetry.
• Latus Rectum: A focal chord perpendicular to the axis of the parabola.
  • Length of the latus rectum: $4a$
  • Ends of the latus rectum: $(a, 2a)$ & $(a, -2a)$

• Perpendicular distance from focus on directrix = half the latus rectum.
• Vertex is the middle point of the focus & the point of intersection of the directrix & axis.
• Two parabolas are said to be equal if they have a latus rectum of the same length.

5. Parametric Representation

The simplest form of representing the coordinates of a point on the parabola $y^2 = 4ax$ is $(at^2, 2at)$. The equations $x = at^2$ & $y = 2at$ together represent the parabola $y^2 = 4ax$, $t$ being the parameter.

6. Type of Parabola

Four standard forms of the parabola:

• $y^2 = 4ax$
• $y^2 = -4ax$
• $x^2 = 4ay$
• $x^2 = -4ay$

7. Position of a Point Relative to a Parabola

The point $(x_1, y_1)$ lies outside, on, or inside the parabola $y^2 = 4ax$ according to the expression $y_1^2 – 4ax_1$ being positive, zero, or negative.

8. Chord Joining Two Points

The equation of a chord of the parabola $y^2 = 4ax$ joining its two points $P(t_1)$ and $Q(t_2)$ is $y(t_1 + t_2) = 2x + 2at_1t_2$.

• If $PQ$ is a focal chord then $t_1t_2 = -1$.
• Extremities of focal chord can be taken as $(at^2, 2at)$ & $(\frac{a}{t^2}, \frac{-2a}{t})$
• If $t_1t_2 = k$ then chord always passes a fixed point (–ka, 0).

9. Line & A Parabola

Tangency Condition:

The line $y = mx + c$ meets the parabola $y^2 = 4ax$ in two points real, coincident, or imaginary according to: $a > =< cm$
$\implies$ Condition of tangency: $c = \frac{a}{m}$

Length of Chord

Length of the chord intercepted by the parabola $y^2 = 4ax$ on the line y = mx + c is : $\frac{4}{m^2} \sqrt{a(1+m^2)(a-mc)}$

• The length of the focal chord, which makes an angle a with the x-axis, is 4a.Cosec$^2 \alpha$.

10. Length of Subtangent & Subnormal

Let $P(at^2, 2at)$ be the point on the parabola $y^2 = 4ax$.

• Length of subtangent: Twice the abscissa of the point $P$.
• Length of subnormal: Constant for all points on the parabola & equal to its semi-latus rectum $(2a)$.

11. Tangent to the Parabola

Point Form:

Equation of tangent at $(x_1, y_1)$: $yy_1 = 2a (x + x_1)$

Slope Form:

Equation of tangent with slope $m$: $y = mx + \frac{a}{m}$, (m $\ne$ 0) Point of contact: $\left(\frac{a}{m^2}, -\frac{2a}{m}\right)$

Parametric form:

The equation of tangent to the given parabola at its point P(t) is - ty = x + at$^2$

• Point of intersection of the tangents at the point $t_1$ & $t_2$ is [$at_1t_2, a(t_1 + t_2)$]. (i.e. G.M. and A.M. of abscissae and ordinates of the points)

12. Normal to the Parabola

Point Form:

Equation of normal at $(x_1, y_1)$: $y – y_1 = -\frac{1}{2a} y_1 (x – x_1)$

Slope Form:

Equation of normal with slope $m$: $y = mx - 2am - am^3$ Foot of the normal: $(am^2, -2am)$

Parametric form:

The equation of normal to the given parabola at its point P(t) is y + tx = 2at + at$^3$

• Point of intersection of normals at $t_1$ & $t_2$ is $a(t_1^2 + t_2^2 + t_1t_2 + 2), -at_1t_2(t_1 + t_2)$
• If the normal to the parabola y$^2$ = 4ax at the point $t_1$, meets the parabola again at the point $t_2$, then $t_2 = -(t_1 + \frac{2}{t_1})$

13. Pair of Tangents

The equation of the pair of tangents from $P(x_1, y_1)$ to the parabola $y^2 = 4ax$ is: $SS_1 = T^2$ Where:

• $S \equiv y^2 - 4ax$
• $S_1 \equiv y_1^2 - 4ax_1$
• $T \equiv yy_1 - 2a (x + x_1)$

14. Chord of Contact

Equation of the chord of contact from $P(x_1, y_1)$: $yy_1 = 2a(x + x_1)$

• Remember that the area of the triangle formed by the tangents from the point $(x_1, y_1)$ & the chord of contact is $\frac{(y_1^2 - 4ax_1)^\frac{3}{2}}{2a}$.
• Also, note that the chord of contact exists only if the point P is not inside.

15. Chord with a Given Middle Point

Equation of the chord of the parabola $y^2 = 4ax$ with middle point $(x_1, y_1)$: $y – y_1 = \frac{2a}{y_1} (x – x_1)$

16. Diameter

The locus of the middle points of a system of parallel chords of a parabola $y^2 = 4ax$ is called a Diameter. Equation to the diameter of a parabola $y^2 = 4ax$: $y = \frac{2a}{m}$ Where $m$ = slope of parallel chords.

17. Co-normal Points

The foot of the normals of three concurrent normals are called Conormal Points. The algebraic sum of the slopes of three concurrent normals is zero.

• The sum of the ordinates of the three normal points is zero.
• The centroid of the triangle is formed by three co-normal points that lie on the axis.
• If $27ak^2 < 4(h – 2a)^3$, then three real and distinct normals can be drawn from point $(h, k)$.
• If three normals are drawn from $(h, 0)$, then $h > 2a$, one normal is the axis, and the other two are equally inclined to the axis.

18. Important Highlights

Tangent and Normal Properties:

• The tangent & normal at any point $P$ intersect the axis at $T$ & $G$. $ST = SG = SP$, where $S$ is the focus.
• Tangent and normal at $P$ bisect the angle between the focal radius $SP$ & the perpendicular from $P$ on the directrix.
• Rays from $S$ become parallel to the axis after reflection.
• Portion of a tangent cut off between the directrix & the curve subtends a right angle at the focus.
• Tangents at the extremities of a focal chord intersect at right angles on the directrix.
• A circle on any focal chord as diameter touches the directrix.
• Circle on any focal radii as diameter touches the tangent at the vertex and intercepts a chord of length $2a (1+t)$ on a normal at $P$.
• Any tangent & the perpendicular from the focus meet on the tangent at the vertex.
• Semi latus rectum is the harmonic mean between segments of any focal chord: $2a = \frac{2bc}{b+c}$
• Image of the focus lies on the directrix with respect to any tangent.

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Circle Formula Sheet
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