Find the coordinates of the focus, the axis of the parabola, the equation of directrix, and the length of the latus rectum for \(y^2 = 12x\).
Find the coordinates of the focus, the axis of the parabola, the equation of directrix, and the length of the latus rectum for \(y^2 = 12x\).
Solution and Explanation
The given equation is \(y^2= 12x. \)
Here, the coefficient of \(x\) is positive.
Hence, the parabola opens towards the right.
On comparing this equation with \(y^2 = 4ax\) , we obtain
\(4a= 12\)
\(⇒ a = 3\)\(\)
∴Coordinates of the focus \(= (a, 0) = (3, 0)\)
Since the given equation involves \(y^2 \), the axis of the parabola is the x-axis.
Equation of direcctrix ,\( x= -a \)
i.e., \(x = - 3\)
\(⇒ x+ 3 = 0 \)
Length of latus rectum : \( 4a= 4*3\) \(=12\)\(\) (Ans.)
Top CBSE Class XI Mathematics Questions
- A coin is tossed twice, what is the probability that atleast one tail occurs?
- The relation f is defined by
\(f(x) = \begin{cases} x^2, & \quad 0≤x≤3\\ 3x, & \quad 3≤x≤10 \end{cases}\)
The relation g is defined by
\(g(x) = \begin{cases} x^2, & \quad 0≤x≤2\\ 3x, & \quad 2≤x≤10 \end{cases}\)
Show that f is a function and g is not a function. - Let \(f={(x,\frac {x^2}{1+x^2}):x∈R}\) be a function from R into R. Determine the range of f.
- How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
- In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together?
Top CBSE Class XI Questions
- The total number of protons in $ 10\,g$ of calcium carbonate is $ (N_{0}=6.023\times 10^{23})$ :
- Justify giving reactions that among halogens, fluorine is the best oxidant and among hydrohalic compounds, hydroiodic acid is the best reductant.
- An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol–1 K–1, molecular mass of O2 = 32 u).
- Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m3 at a temperature of 27 °C and 1 atm pressure.
- An air bubble of volume 1.0 cm3 rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C ?
Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.