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Parabola, Length of the Latus rectum and parametric form, Other Form of Parabola, General equation of Parabola is considered one of the most asked concept.
114 Questions around this concept.
The locus of the centre of a circle which touches a given line and passes through a given point, not lying on the given line, is
Equation of parabola having it's vertex at and focus at is
Vertex of the parabola whose parametric equation is , is
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Equation of parabola having it's focus at and one extremity of it's latus rectum as is
A point on the parabola $y^2=18 x$ at which the ordinate increases at twice the rate of the abscissa is
Equation of parabola having the extremities of it's latus rectum as and is
The centres of those circles which touch the circle, $x^2+y^2-8 x-8 y-4=0$, externally and also touch the $x$ axis, lie on :
Parabola
A parabola is the locus of a point moving in a plane such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).
Hence it is a conic section with eccentricity $\mathrm{e}=1$.
$
\begin{aligned}
& \frac{P S}{P M}=e=1 \\
& \Rightarrow P S=P M
\end{aligned}
$
Standard equation of a parabola
Let focus of parabola is $S(a, 0)$ and directrix be $x+a=0$
$P(x, y)$ is any point on the parabola.
Now, from the definition of the parabola,
$
\begin{array}{cc}
& \mathrm{SP}=\mathrm{PM} \\
\Rightarrow & \mathrm{SP}^2=\mathrm{PM}^2 \\
\Rightarrow & (\mathrm{x}-\mathrm{a})^2+(\mathrm{y}-0)^2=(\mathrm{x}+\mathrm{a})^2 \\
\Rightarrow & \mathrm{y}^2=4 \mathrm{ax}
\end{array}
$
Which is the required equation of a standard parabola
Important Terms related to Parabola
1. Axis: The line which passes through the focus and is perpendicular to the Directrix of the parabola. For the parabola $\mathbf{y}^2=\mathbf{4 an x}$, the X-axis is the Axis.
2. Vertex: The point of intersection of the parabola and axis. For parabola $\mathbf{y}^{\mathbf{2}}=\mathbf{4 a x}, \mathrm{A}(0,0)$ i.e. origin is the Vertex.
3. Double Ordinate: Suppose a line perpendicular to the axis of the parabola meets the curve at Q and $\mathrm{Q}^{\prime}$. Then, $\mathrm{QQ}^{\prime}$ ' is called the double ordinate of the parabola.
4. Latus Rectum: The double ordinate LL' passing through the focus is called the latus rectum of the parabola.
5. Focal Chord: A chord of a parabola which is passing through the focus. In the figure PP' and LL' are the focal chord.
6. Focal Distance: The distance from the focus to any point on the parabola. I.e. PS
$\mathrm{SP}=\mathrm{PM}=$ Distance of P from the directrix
$P=(x, y)$
$\mathrm{SP}=\mathrm{PM}=\mathrm{x}+\mathrm{a}$
Length of the Latus rectum and Parametric form
The length of the latus rectum of the parabola $y^2=4 a x_{\text {is }} 4 \mathrm{a}$.
L and $\mathrm{L}^{\prime}$ be the ends of the latus rectum.
The $x$-coordinate of $L$ and $L$ ' will be ' $a$ '.
As $L$ and $L^{\prime}$ lie on parabola, so put $x=a$ in the equation of parabola to get $y$ coordinates of $L$ and $L^{\prime}$
$
\begin{aligned}
& y^2=4 a \cdot a=4 a^2 \\
& \Rightarrow y= \pm 2 a \\
& \therefore L(a, 2 a) \text { and } L^{\prime}(a,-2 a) \\
& \text { so that } L L^{\prime}=4 a
\end{aligned}
$
Parametric Equation:
From the equation of the parabola, we can write $\frac{y}{2 a}=\frac{2 x}{y}=t$ here, t is a parameter
Then, $x=a t^2$ and $y=2 a t$ are called the parametric equations and the point $\left(a t^2, 2 a t\right)$ lies on the parabola.
Point $\mathbf{P}(\mathbf{t})$ lying on the parabola means the coordinates of $P$ are (at ${ }^{\mathbf{2}}, \mathbf{2 a t}$ )
Other Forms of Parabola
1.Parabola with focus $S(-a, 0)$ and directrix $x=a$ (Parabola opening to left)
Equation of the parabola is $\mathrm{y}^2=-4 \mathrm{ax}, \mathrm{a}>0$
2.Parabola with focus $S(0, a)$ and directrix $y=-a$ (Parabola opening to upward)
The equation of the parabola is $x^2=4 a y, a>0$.
3.Parabola with focus $S(0,-a)$ and directrix $y=a$ (Parabola opening to downward)
The equation of the parabola is $x^2=-4 a y, a>0$.
General equation of Parabola
Let $S(h, k)$ be the focus $I x+m y+n=0$ be the equation of the directrix and $P(x, y)$ be any point on the parabola.
Then, from the definition PS = PM
$
\Rightarrow \quad \sqrt{(x-h)^2+(y-k)^2}=\left|\frac{l x+m y+n}{\sqrt{\left(l^2+m^2\right)}}\right|
$
Squaring both sides, we get
$
\Rightarrow \quad(x-h)^2+(y-k)^2=\frac{(l x+m y+n)^2}{\left(l^2+m^2\right)}
$
This is the general equation of a parabola.
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