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Lesson Posted on 07 Feb Conic Sections

Why Study Ellipse?

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

Why study ellipses? Orbiting satellites (including the earth revolving around the sun, and the moon revolving around us) trace out elliptical path. Many buildings and bridges use the ellipse as a pleasing (and strong) shape. One... read more

Why study ellipses?

Orbiting satellites (including the earth revolving around the sun, and the moon revolving around us) trace out elliptical path.

                           earth revolving around sun

Many buildings and bridges use the ellipse as a pleasing (and strong) shape.

                                           elliptical bridge

One property of ellipses is that a sound (or any radiation) beginning in one focus of the ellipse will be reflected so it can be heard clearly at the other focus.

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Lesson Posted on 07 Feb Conic Sections

Equation Of Ellipse

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

Ellipses with Horizontal Major Axis: Equation for an ellipse with a horizontal major axis is given by: read more

Ellipses with Horizontal Major Axis:

Equation for an ellipse with a horizontal major axis is given by:

                        

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Lesson Posted on 07 Feb Conic Sections

Ellipse As Locus

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

The ellipse is defined as the locus of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses) is constant. We can produce an ellipse by pinning the ends of a piece of string and keeping a pencil tightly within the boundary of the string, as follows. We... read more

The ellipse is defined as the locus of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses) is constant.

We can produce an ellipse by pinning the ends of a piece of string and keeping a pencil tightly within the boundary of the string, as follows.

We start with these 2 foci:

                                                          pencil ellipse

We pin the ends of the string to the foci and begin to draw, holding the string tight:

                                                    pencil ellipse

                                                   pencil ellipse

                                                 pencil ellipse

                                                pencil ellipse

Our complete ellipse is formed:

                                                 pencil ellipse

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Lesson Posted on 07 Feb Conic Sections

Why We Study Hyperbola?

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane. How can we obtain a hyperbola from slicing a cone? We start with a double cone (2 right circular cones placed apex to apex): ... read more

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

How can we obtain a hyperbola from slicing a cone?

We start with a double cone (2 right circular cones placed apex to apex):

                                                    double cone

When we slice the 2 cones vertically, we get a hyperbola, as shown:                                                                                                                                                                                                         hyperbola conic section

                                                  hyperbolic cooling towers

Cooling towers for a nuclear power plant have a hyperbolic cross-section

How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) = 2a units.

The curve that results is called a hyperbola. There are two parts to the curve.

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Lesson Posted on 06 Feb Conic Sections

Hyperbola And It's Summary

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane. How do we create a hyperbola? Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units). Now, move along a curve... read more

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) = 2a Units

                                                                  hyperbolic cooling towers

Cooling towers for a nuclear power plant have a hyperbolic cross-section

General Equation of North-South Hyperbola:

For the hyperbola with focal distance 4a (distance between the 2 foci), and passing through the y-axis at (0, c) and (0, −c), we defineCooling towers for a nuclear power plant have a hyperbolic cross-section

    b2 = c2 − a2

Applying the distance formula for the general case, in a similar fashion to the above example, we obtain the general form for a north-south hyperbola:

    y2/a2-x2/b2 = 1

 

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Lesson Posted on 03 Feb Conic Sections

Summary Of Circle & Relation With Conic section

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

If we slice one of the cones with a plane at right angles to the axis of the cone, the shape formed is a circle. The circle with centre (0, 0) and radius r has the equation: x2 + y2... read more

                                                                   conic - circle

If we slice one of the cones with a plane at right angles to the axis of the cone, the shape formed is a circle.

                                             circle center origincircle center (h,k)

 

The circle with centre (0, 0) and radius r has the equation: x2 + y2 = r2

The circle with centre (h, k) and radius r has the equation: (x − h)2 + (y − k)2 = r2
 

General Form of the Circle: An equation which can be written in the following form (with constants D, E, F) represents a circle: x2 + y2 + Dx + Ey + F = 0x2 + y2 + 2gx + 2fy + c = 0Radius=√g2+f2-ccenter of Circle is given by: -g,-f 
  

 
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Lesson Posted on 03 Feb Conic Sections

Locus & Its Example

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

The set of all points that share a property. This usually results in a curve or surface. Example: Circle: A Circle is "the locus of points on a plane that are a certain distance from a central point". A circle is the locus of points that are equidistant... read more

 

                                        Image result for example of locus in maths

 

The set of all points that share a property. This usually results in a curve or surface.

Example:

Circle:

  • A Circle is "the locus of points on a plane that are a certain distance from a central point".
  • A circle is the locus of points that are equidistant from a fixed point (the center).

Parabola:

  • A parabola is the locus of points that are equidistant from a point (the focus) and a line (the directrix).

Ellipse:

  •  An ellipse is the locus of points whereby the sum of the distances from 2 fixed points (the foci) is constant. 
Hyperbola:
 
  • A hyperbola is the locus of points where the difference in the distance to two fixed foci is constant.

                         Image result for example of locus

 

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Lesson Posted on 03 Feb Conic Sections

Summary Of Straight Line & Relation With Conic Section

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

1. Slope-Intercept form: Y=mx+b 2. Point-slope form: The equation of a line passing through a point (x1, y1) with slope m y − y1 = m(x −... read more

 

1. Slope-Intercept form:

 

                                                                       y = mx + b graph

Y=mx+b

 

2. Point-slope form:

                                                                   point-slope graph

The equation of a line passing through a point (x1, y1) with slope m

y − y1 = m(x − x1)

Genreral Form of Straight line is given by:

Ax+By+C=0

                                                                     conic - straight line

 

Conic Section:

If we slice the double cone by a plane just touching one edge of the double cone, the intersection is a straight line, as shown above.

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Answered on 03/12/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Conic Sections

Show that the point (5, 6) outside the circle: (square of x) + (square of y) = 4?

Sarvajeet Kumar

An Experienced Trainer

S'=(square of 5) + (square of 6) - 4 = -57.
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Answered on 03/12/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Conic Sections

Find the equation of the circle which passes through the points (1, 2), (2, 2) and (4, -1)?

Sarvajeet Kumar

An Experienced Trainer

square of (x-3/2) + square of (y-3/2) = 1/2.
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