## Conic Sections

# Parabola

**Tips :**

- Equation of a conic with (x
_{1}y_{1}) focus lx + my + n = 0 directrix and eccentricity ‘e’ is (l^{2}+ m^{2}) [(x - x_{1})^{2}+ (y - y_{1})^{2}] = e^{2}(lx + my + n)^{2 } - A parabola is the locus of a point which moves such that its distance from a fixed point is always equal to its distance from a fixed straight line.
- S = 0 will represent pair of straight lines if \tt \begin{vmatrix}a & h & g \\h & b & f \\ g & f & c\end{vmatrix}=0
- If Δ ≠ 0, h
^{2}= ab then s = 0 will represent parabola. - The equation (y − k)
^{2}= 4a (x – h) represents a parabola with axis y − k = 0, vertex (h, k) focus (h + a, k), directrix x = h – a and length of latus rectum 4a. - If t
_{1}and t_{2}are the extremities of a focal chord of a parabola then t_{1}t_{2}= -1 - Equation of the tangent to y
^{2}= 4ax at (x_{1}y_{1}) is y y_{1}= 2a (x + x_{1}) - Equation of the tangent y
^{2}= 4ax at the point ‘t’ is yt = x + at^{2} - Condition of Tangency for the parabola y
^{2}= 4ax is \tt c= \frac{a}{m}. Here the line is y = mx + c and the point of contact is \tt \left(\frac{a}{m^{2}},\frac{2a}{m}\right) - Tangent to (y - k)
^{2}= ± 4a (x – h) of slope ‘m’ will be the form y - k = m (x – h) ± \tt \frac{a}{m} - Tangent to (x – h)
^{2}= ± 4a (y – k) of slope ‘m’ will be the form y - k = m (x – h) \tt \mp \ am^{2} - The point of intersection of the tangents at t
_{1}, t_{2}on the parabola y^{2}= 4ax is [at_{1}t_{2}, a(t_{1}+ t_{2})] - The tangents at the ends of a focal chord of the parabola meet on the directrix at right angles.
- The tangent at one end of a focal chord of a parabola is parallel to the normal at the other end.
- The locus of the foot of the perpendicular from the focus to the tangent of a parabola is the tangent at the vertex.
- Equation of common tangent to the parabolas y
^{2}= 4ax and x^{2}= 4by is \tt xa^{\frac{1}{3}}+yb^{\frac{1}{3}}+a^{\frac{2}{3}}.b^{\frac{2}{3}}=0 - The maximum number of common normal to y
^{2}= 4ax and x^{2}= 4by may be ‘5’ only. - Area of the triangle formed by the tangents from (x
_{1}y_{1}) to the parabola y^{2}= 4ax and its chord of contact is \tt \frac{\left(y_1^2-4a\ x_{1}\right)^{3/2}}{2a} - The length of chord of contact of tangents drawn from (x
_{1}y_{1}) to the parabola y^{2}= 4ax is \tt \frac{\sqrt{\left(y_1^2-4a x_{1}\right)\left(y_1^2+4a^{2}\right)}}{a} - For any conic y = f(x), length of the chord made by y = mx + c is \tt \sqrt{m^{2}+4c}.\sqrt{1+m^{2}} where ‘m’ is slope of the line.
- From an external point two tangents can be drawn to a parabola. The slopes of the two tangents to y
^{2}= 4ax passing through (x_{1}y_{1}) are given by the equation m^{2}x_{1}– m y_{1}+ a = 0 - If m
_{1}, m_{2}be the slopes of the two tangents drawn from (x_{1}y_{1}) to y^{2}= 4ax then \tt m_{1}+m_{2}=\frac{y_{1}}{x_{1}},\ m_{1}\ m_{2}=\frac{a}{x_{1}} - If 'θ' is the angle between the pair of tangents drawn from (x
_{1}y_{1}) to y^{2}= 4ax then \tt \tan\theta = \frac{\sqrt{y_1^2-4\ ax_{1}}}{x_{1}+a}\ (or)\ \frac{\sqrt{S_{11}}}{x_{1}+a} - The locus of point of intersection of the two tangents to y
^{2}= 4ax included at a constant angle ‘α’ is (x + a)^{2}tan^{2}α = y^{2}– 4ax - The condition for lx + my + n = 0 to be a normal to y
^{2}= 4ax is al^{3}+ 2al m^{2}+ m^{2}n = 0 - Equation of the normal to y
^{2}= 4ax having slope ‘m’ is y = mx – 2am – am^{3}and the foot of the normal is (am^{2}, -2am) - The centroid of the triangle formed by the foot of the ‘3’ normal of the parabola lies on the axis of the parabola.
- If the lines y = mx + c intersect the parabola y
^{2}= 4ax in the points A and B then the length of the chord AB is \tt \frac{4}{m^{2}}\sqrt{a\left(a-mc\right)\left(1+m^{2}\right)} and the midpoint of the chord AB is \tt \left(\frac{2a-mc}{m^{2}},\frac{2a}{m}\right) - If (at
^{2}, 2at) is one end of a focal chord of the parabola then the other end is \tt \left(\frac{a}{t^{2}},\frac{-2a}{t}\right) - The length of the focal chord at ‘t’ is \tt a\left(t+\frac{1}{t}\right)^2
- If PSQ is a focal chord of the parabola y
^{2}= 4ax with focus ‘S’ then \tt \frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a} - The least length of a focal chord of a parabola is its length of latusrectum.
- Length of the focal chord of y
^{2}= 4ax making an angle θ with its axis is 4a cosec^{2}θ - Length of the chord of the parabola y
^{2}= 4ax passing through the vertex and making an angle ‘θ’ with its axis is 4a cos θ cosec^{2}θ - The (x
_{1}y_{1}) (x_{2}y_{2}) are the extremities of a focal chord of the parabola y^{2}= 4ax then x_{1}x_{2}= a^{2}; y_{1}y_{2}= -4a^{2} - The orthocentre of the triangle formed by the tangents at t
_{1}, t_{2}, t_{3}to the parabola y^{2}= 4ax is (-a, a t_{1}+t_{2}+t_{3}+ t_{1}t_{2}t_{3}) - Area of triangle inscribed in parabola y
^{2}= 4ax is \tt \frac{1}{8a}\mid\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\mid where y_{1}, y_{2}and y_{3}are ordinates of angular points. - Area of triangle formed by tangents drawn to the parabola y
^{2}= 4ax at the points whose ordinates are y_{1}, y_{2}and y_{3}is \tt \frac{1}{16a}\mid\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\mid - The length of the chord joining t
_{1}, t_{2}on y^{2}= 4ax is \tt a \mid t_{1}-t_{2}\mid\sqrt{\left(t_{1}+t_{2}\right)^2+4} - If the normal at ‘t’ on the parabola y
^{2}= 4ax subtends a right angle at its focus then t = ± 2 as its vertex then \tt t =\pm\ \sqrt{2}

### Equation of Parabola

### Latus Rectum in Parabola,Examples

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1. Standard form of the parabola is y^{2} = 4ax.

2. Parabola opening to left i.e, y^{2} = −4ax

3. Parabola opening upwards i.e., x^{2} = 4ay

4. Parabola opening downwards i.e., x^{2} = −4ay

5.

Important terms |
y^{2 }= 4ax |
y^{2 }= −4ax |
x^{2 }= 4ay |
x^{2} = -4ay |

Vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |

Focus | (a, 0) | (−a, 0) | (0, a) | (0, −a) |

Directrix | x = − a | x = a | y = − a | y = a |

Axis | y = 0 | y = 0 | x = 0 | x = 0 |

Latustrectum | 4a | 4a | 4a | 4a |

Focal distance P(x, y) | x + a | x − a | y + a | a − y |

6. If the vertex of the parabola is at a point A(h, k) and its latustrectum is of length 4a, then its equation is

a) (y − k)^{2} = 4a (x − h), its axis is parallel to OX i.e., parabola open rightward.

b) (y − k)^{2} = − 4a (x − h), its axis is parallel to OX' i.e., parabola open leftward.

c) (x − h)^{2} = − 4a (y − k), its axis is parallel to OY i.e., parabola open upward.

d) (x − h)^{2} = − 4a (y − k), its axis is parallel to OY' i.e., parabola open downward.

7. **Point Form**

The equation of the tangent to the parabola y^{2} = 4ax at a point (x_{1} , y_{1}) is given by yy_{1}= 2a (x + x_{1}).

8. **Slope Form**

a) The equation of the tangent of slope m to the parabola y^{2} = 4ax is \tt y=mx+\frac{a}{m}

b) The equation of the tangent of slope m to the parabola (y − k)^{2} = 4a (x − h) is given by (y − k) = m (x − h) + \tt \frac{a}{m}

c) The coordinates of the point of contact are \tt \left(h+\frac{a}{m^{2}},k+\frac{2a}{m}\right)

9. **Parametric Form: **The equation of the tangent to the parabola y^{2} = 4ax at a point (at^{2} , 2at) is yt = x + at^{2}.

10. **Point Form: **The equation of the normal to the parabola y^{2} = 4ax at a point (x_{1} , y_{1}) is given by y − y_{1} = \tt -\frac{y_{1}}{2a}\left(x-x_{1}\right).

11. **Parametric Form : **The equation of the normal to the parabola y^{2} = 4ax at point (at^{2} , 2at) is given by y + tx = 2at + at^{3}.

12. **Slope Form: **The equation of the normal to the parabola y^{2} = 4ax in terms of its slope m is given by y = mx − 2am − am^{3} at point (am^{2} , −2am).