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Lesson Posted 3 days ago Co-ordinate Geometry

Tangent and normal ,Subtangent,Subnormal

Raj Kumar

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

Equation of tangent at P(x,y) to the curve y=f(x) is given as Y-y=dy/dx(X-x) Equation of Normal at P(x,y) to the curve y=f(x) is given as Y-y=-1/dy/dx(X-x) By applying the condition of perpendicularity If product of slopes of lines will be equal to -1 lines are perpendicular as we know that... read more

Image result for tangent normal subtangent subnormal

Equation of tangent at P(x,y) to the curve y=f(x) is given as

Y-y=dy/dx(X-x)

Equation of Normal  at P(x,y) to the curve y=f(x) is given as

Y-y=-1/dy/dx(X-x)

By applying the condition of perpendicularity

If product of slopes of lines will be equal to -1 lines are perpendicular

as we know that normal and tangent will be perpendicular to each other

m1m2=-1

let slope of tangent is m1=dy/dx

so solpe of normal will be m2=-1/m1=-1/dy/dx

length of tangnet will be y=√1+(dx/dy)2

Length of normal will be y=√1+(dy/dx)2

length of Sub-Tangent will be ydx/dy

length of Sub-normal  will be ydy/dx

 

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Lesson Posted 3 days ago Co-ordinate Geometry

Ellipse

Raj Kumar

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

Ellipse is a regular oval shape, traced by a point moving in a plane so that the the sum of its distances from two other points (the foci) is constant When a cone is cut by an oblique plane which does not intersect the base, it results in the formation of an ellipse. Standard equation of the ellipse... read more

Ellipse is a regular oval shape, traced by a point moving in a plane so that the

the sum of its distances from two other points (the foci) is constant

When a cone is cut by an oblique plane which does not intersect the base, it results in the formation of an ellipse.

Standard equation of the ellipse is given as

x2/a2+y2/b2=1

and eccentricity of the ellipse lies between 0 and 1

the eccentricity of an ellipse is given as

e=√1-b2/a2 when a>b

Foci of an ellipse are (±ae,0)

Major Axis=2a

Minor Axis=2b

The equation of Directrix are x=±a/e

Coordinates of vertices are (±a,0)

Length of latus rectum is 2b2/a.

 

 

Ellipse

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Lesson Posted on 03 Jan Co-ordinate Geometry

Graphical Transformation

Raj Kumar

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

Graphical Transformation: read more

 Graphical Transformation:

https://www.onlinemathlearning.com/image-files/transformation-rules-graphs.png

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Lesson Posted on 03 Jan Co-ordinate Geometry

Gradient (Slope) Of Line / Inclination

Raj Kumar

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

Gradient is defined as inclination or declination Slope/Gradient = Verticle Rise/HorizontaL Run In the following triangle, the gradient of the line is given by: i. Slope = m = (y2-y1/X2-X1) = Change in Y/Chane in X ii. Slope = Verticle Rise/HorizontaL Run iii. Slope = Change in absicss/Change in... read more

Gradient is defined as inclination or declination

Slope/Gradient = Verticle Rise/HorizontaL Run

In the following triangle, the gradient of the line is given by:

i. Slope = m = (y2-y1/X2-X1) = Change in Y/Chane in X

ii. Slope = Verticle Rise/HorizontaL Run

iii. Slope = Change in absicss/Change in Ordinate

iv. Slope = tanθ = a/b = Tangent at specific point = inclination or delination.

                                       slope of a line diagram   

                                              right triangle

Positive & negative Slopes: A positive slope indicates the value of the dependent variable increases as we go left to right.

                                              graph of line with positive slope

Negative Slope: A negative slope means that the value of the dependent variable (usually y) is decreasing as we go left to right:

                                              graph of line with negative slope

Inclination:

                                              graph of inclination

We have a line with slope m and the angle that the line makes with the x-axis is α.

From trigonometry, we recall that the tan of angle α is given by:

tan α = opposite/adjacent

                                            graph of inclination

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Lesson Posted on 27/12/2017 Co-ordinate Geometry

Geometrical Transformation

Raj Kumar

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

Transformation is basically the "change of form" Etymology Trans = across, bayond + formare = to form These transformations fall into two categories: i. Rigid transformations that do not change the shape or size of the pre-image ii. Non-rigid transformations that change the size but not the shape... read more

Transformation is basically the "change of form"

Etymology Trans = across, bayond  + formare = to form

TransformationThese transformations fall into two categories:

i. Rigid transformations that do not change the shape or size of the pre-image

ii. Non-rigid transformations that change the size but not the shape of the pre-image.

 

Basically four type of transformation:

1) Translation (Shifting/Sliding in linear direction) or Horizontal or vertical movement either upward or downward 

 

2) Rotation:

i. A rotation is atransformation in which the object is rotated about a fixed point.

ii. Turn Around either in Clockwise or in anticlockwise/Counterclockwise direction

Reflection can be:

a. Around X axis

b. Around Y Axis

c. Around line Y=X or Y=-X

d. Around the Origin

 

3) Reflection:

i. Mirror Image of anything

ii. It is basically a 'flip' of a shape over the line of reflection

 

4) Dilation:

i. Dilation that creates a smaller image is called a reduction.

ii. A dilation stretches or shrinks the original figure.

iii. A description of a dilation includes the scale factor (or ratio) and the center of the dilation.

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Lesson Posted on 27/12/2017 Co-ordinate Geometry

Geometry

Raj Kumar

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

Important Properties of Paralleolgram: 1) Opposite side length are Exactly Equal or Congruent that means. 2) Both line pairs will be parallel to each other. 3) Opposite parallel lines slopes are equal. 4) Area of paralellogram = Base X Altitude. 5) Opposite angles are exactly equal or congruent.... read more

Important Properties of Paralleolgram:

Parallelogram Similarity with Square,Rectangle,Rhombus & Trapezium

1) Opposite side length are Exactly Equal or Congruent that means.

2) Both line pairs will be parallel to each other.

3) Opposite parallel lines slopes are equal.

4) Area of paralellogram = Base X Altitude.

5) Opposite angles  are exactly equal or congruent.

6) Sum of adjacent or consective angles is 180°.

7) If one angle is 90° then all angles will be 90°.

8) Diagnols of parallelogram bisect each other.

9) Each diagonal of parallelogram seperates into two exactly equal triangles.

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Answered on 14/12/2016 CBSE/Class 10/Mathematics Tuition/Class IX-X Tuition Co-ordinate Geometry

What do you mean by abscissa and ordinate of a given point?

Ravikumar

Abscissa. The horizontal ("x") value in a pair of coordinates. The ordinate of an ordered pair (x, y) is its second component, i.e. it is y.
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Answered on 14/12/2016 CBSE/Class 10/Mathematics Tuition/Class IX-X Tuition Co-ordinate Geometry

What is the mid-point formula of a line AB with a mid-point C?

Ravikumar

Suppose A=(x1,y1) and B=(x2,y2) are the end points of a line segment AB Then the mid point C is given by: C =.
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Answered on 14/12/2016 CBSE/Class 10/Mathematics Tuition/Class IX-X Tuition Co-ordinate Geometry

What do you mean by Cartesian plane?

Ravikumar

It is a plane which has a horizontal line (x axis) and a vertical line (y axis). It is also known as coordinate plane.
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Answered on 14/12/2016 CBSE/Class 10/Mathematics Tuition/Class IX-X Tuition Co-ordinate Geometry

Does the point (-3, -5) lies on the y-axis?

Ravikumar

No, It lies in the III quadrant.
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