Summary
Summary Concepts of Modern Mathematics, ISBN: 9780486134956 Engineering maths
- Course
- Institution
- Book
Summary of Parabola. A complete handbook.
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
Seller
Follow
gayatriarya
Content preview
ParabolaConic Section
A conic is the locus of a point whose distance from a fixed point bears a
constant ratio to its distance from a fixed line. The fixed point is the
focus S and the fixed line is directrix l.
Z
P
M
directrix
S
(focus)
Z′
The constant ratio is called the eccentricity denoted by e.
(i) If 0 < e < 1, conic is an ellipse.
(ii) e = 1, conic is a parabola.
(iii) e > 1, conic is a hyperbola.
General Equation of Conic
If fixed point of curve is ( x1 , y1 ) and fixed line is ax + by + c = 0, then
equation of the conic is
( a 2 + b2 ) [( x − x1 )2 + ( y − y1 )2 ] = e2( ax + by + c)2
which on simplification takes the form
ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0,
where a , b, c, f , g and h are constants.
A second degree equation ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0
represents
a h g
(i) a pair of straight lines, if ∆ = h b f = 0
g f c
, (ii) a pair of parallel (or coincident) straight lines, if ∆ = 0
and h 2 = ab.
(iii) a pair of perpendicular straight lines, if ∆ = 0 and a + b = 0
(iv) a point, if ∆ = 0 and h 2 < ab
(v) a circle, if a = b ≠ 0, h = 0 and ∆ ≠ 0
(vi) a parabola, if h 2 = ab and ∆ ≠ 0
(vii) a ellipse, if h 2 < ab and ∆ ≠ 0
(viii) a hyperbola, if h 2 > ab and ∆ ≠ 0
(ix) a rectangular hyperbola, if h 2 > ab, a + b = 0 and ∆ ≠ 0
Important Terms Related to Parabola
(i) Axis A line perpendicular to the directrix and passes through
the focus.
(ii) Vertex The intersection point of the conic and axis.
(iii) Centre The point which bisects every chord of the conic passing
through it.
(iv) Focal Chord Any chord passing through the focus.
(v) Double Ordinate A chord perpendicular to the axis of a conic.
(vi) Latusrectum A double ordinate passing through the focus of
the parabola.
(vii) Focal Distance The distance of a point P ( x , y ) from the focus S
is called the focal distance of the point P.
Parabola
A parabola is the locus of a point which moves in a plane such that its
distance from a fixed point in the plane is always equal to its distance
from a fixed straight line in the same plane.
If focus of a parabola is S ( x1 , y1 ) and equation of the directrix is
ax + by + c = 0, then the equation of the parabola is
( a 2 + b2 )[( x − x1 )2 + ( y − y1 )2 ] = ( ax + by + c)2
Y
, y)
P(x
X' X
O S(x1, y1)
ax + by + c = 0
Y'
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller gayatriarya. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
77973 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now