OP Malhotra Class-11 Parabola S.Chand ISC Maths Solution Chapter-23. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-23, With Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.
OP Malhotra Class-11 Parabola S.Chand ISC Maths Solution
| Class: | 11th |
| Subject: | Mathematics |
| Chapter : | Ch-23 Parabola of Section -A |
| Board | ISC |
| Writer | OP Malhotra |
| Publications | S.Chand Publications 2020-21 |
–: Select Topics :-
OP Malhotra Class-11 Parabola S.Chand ISC Maths Solution
Parabola
Section of a right circular cone by a plane parallel to a generator of the cone is a parabola. It is a locus of a point, which moves so that distance from a fixed point (focus) is equal to the distance from a fixed line (directrix)
- Fixed point is called focus
- Fixed line is called directrix
Standard Equation of Parabola
The simplest equation of a parabola is y2 = x when the directrix is parallel to the y-axis. In general, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as:
| y2 = 4ax |
If the parabola is sideways i.e., the directrix is parallel to x-axis, the standard equation of a parabole becomes,
| x2 = 4ay |
Apart from these two, the equation of a parabola can also be y2 = 4ax and x2 = 4ay if the parabola is in the negative quadrants. Thus, the four equations of a parabola are given as:
- y2 = 4ax
- y2 = – 4ax
- x2 = 4ay
- x2 = – 4ay
Parametric co-ordinates of Parabola:
For a parabola, the equation is y2 = -4ax. Now, to represent the co-ordinates of a point on the parabola, the easiest form will be = at2 and y = 2at as for any value of “t”, the coordinates (at2, 2at) will always satisfy the parabola equation i.e. y2 = 4ax. So,
Any point on the parabola
y2 = 4ax (at2, 2at)
Parabola
A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distance from a fixed line l in the plane. The fixed point F is called focus and the fixed line l is the directrix of the parabola.
Main Facts About the Parabola
| Forms of parabola | y2= 4ax | y2 = -4ax | x2 = 4ay | x2 = -4ay |
| Axis of parabola | y = 0 | y = 0 | x = 0 | x = 0 |
| Directrix of parabola | x = -a | x = a | y = -a | y = a |
| Vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |
| Focus | (a, 0) | (-a, 0) | (0, a) | (0, -a) |
| Length of latus rectum | 4a | 4a | 4a | 4a |
| Focal length | |x + a| | |x – a| | |y + a| | |y – a| |
Parametric Equations of the Parabola y2 = 4ax
The parametric equations of the parabola y2 = 4ax are x = at2 , y = 2at , where I is the parameter. Since the point (at2, 2at) satisfies the equation y2 = 4ax, therefore the parametric coordinates of any point on the parabola are (at2, 2at).
Also, the point (at2 , 2at) is referred to as the parametric point on the parabola.
Relation between a parabola and a line
Let the parabola be y2 = 4ax and the given line be y = mx + c. Solving the line and parabola, we get
(mx + C)2 = 4ax
i.e., m2x2 + 2(cm-2a) x + c2 = 0
Above equation being a quadratic in x
Since, the discriminant = 4 (cm – 2a)2 – 4m2 c = 16a (a-mc)
Hence the line intersects the parabola in 2 distinct points if a > mc , in one point if a = mc , and does not intersect if a < cm.
y = mx + a/m touches the parabola.
Hence, y=mx+ a/m, m≠0 touches the parabola y2 = 4ax at ( a/m2 , 2a/m )
Exe-23
OP Malhotra Class-11 Parabola S.Chand ISC Maths Solution
Page 23-17 to 23-18
Question 1:
Find the equation parabola in problem …………
Question 2:
……………………
…………………….
……………………..
Question 24:
The length of the ……………
(a) 2
(b) 1
(c) 4
(d) None of these
Chapter Test
OP Malhotra Class-11 Parabola S.Chand ISC Maths Solution
Page 23-20
-: End of Parabola Solution :-
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