Learn Co-ordinate Geometry with Free Lessons & Tips

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.

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}

 Graphical Transformation:

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 = (y_2-y_{1/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.

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

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

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

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 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.

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.

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.