Parabola:
Parabola:
Let us learn about the meaning of a Parabola,and also let us learn to solve the equation of a Parabola.
The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola. That is a parabola is a conic whose eccentricity is 1.
Standard equation of a parabola :
Given :
Fixed point (F)
Fixed line (l)
Eccentricity (e = 1)
Moving point P(x, y)
Construction :
Plot the fixed point F and draw the fixed line ‘l’.
Drop a perpendicular (FZ) from F to l.
Take FZ = 2a and treat it as x-axis.
Draw a perpendicular bisector to FZ and treat it as y-axis.
Let V(0, 0) be the origin.
Drop a perpendicular (PM) from P to l.
The known points are F(a, 0), Z(− a, 0) and hence M is (− a, y).
By the definition of a conic,
FP / PM = e = 1 ⇒ FP2 = PM2
(x − a)2 + (y − 0)2 = (x + a)2 + (y − y)2
x2 − 2ax + a2 + y2 = x2 + 2ax + a2 which simplifies to y2 = 4ax.
This is the standard equation of the parabola.
Ellipse:
An ellipse is a conic obtained on slicing across obliquely one nappe of a cone.If P is any point on the ellipse and
F1and F2 its foci, the angle subtended by F1P and F2P with the tangent at P are equal and if a source of light or sound is
placed at one focus of an ellipsoidal reflector (surface generated by revolving an ellipse about its major axis) all the
waves will be reflected so as to pass through the other focus
Hyperbola:
A hyperbola is a conic obtained on slicing a double napped cone by a plane parallel to the axis of the cone. The lines from the foci to any point of a hyperbola make equal angles with the tangent at that point. Hence if the surface of a reflector is generated by revolving a hyperbola about its transverse axis, all rays of light converging on one focus are reflected to the other.
Hope you like the above example of Parabola.Please leave your comments, if you have any doubts.
Parabola:
Let us learn about the meaning of a Parabola,and also let us learn to solve the equation of a Parabola.
The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola. That is a parabola is a conic whose eccentricity is 1.
Standard equation of a parabola :
Given :
Fixed point (F)
Fixed line (l)
Eccentricity (e = 1)
Moving point P(x, y)
Construction :
Plot the fixed point F and draw the fixed line ‘l’.
Drop a perpendicular (FZ) from F to l.
Take FZ = 2a and treat it as x-axis.
Draw a perpendicular bisector to FZ and treat it as y-axis.
Let V(0, 0) be the origin.
Drop a perpendicular (PM) from P to l.
The known points are F(a, 0), Z(− a, 0) and hence M is (− a, y).
By the definition of a conic,
FP / PM = e = 1 ⇒ FP2 = PM2
(x − a)2 + (y − 0)2 = (x + a)2 + (y − y)2
x2 − 2ax + a2 + y2 = x2 + 2ax + a2 which simplifies to y2 = 4ax.
This is the standard equation of the parabola.
Ellipse:
An ellipse is a conic obtained on slicing across obliquely one nappe of a cone.If P is any point on the ellipse and
F1and F2 its foci, the angle subtended by F1P and F2P with the tangent at P are equal and if a source of light or sound is
placed at one focus of an ellipsoidal reflector (surface generated by revolving an ellipse about its major axis) all the
waves will be reflected so as to pass through the other focus
Hyperbola:
A hyperbola is a conic obtained on slicing a double napped cone by a plane parallel to the axis of the cone. The lines from the foci to any point of a hyperbola make equal angles with the tangent at that point. Hence if the surface of a reflector is generated by revolving a hyperbola about its transverse axis, all rays of light converging on one focus are reflected to the other.
Hope you like the above example of Parabola.Please leave your comments, if you have any doubts.
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