Compare general equations expended form only of a circle par

Compare general equations (expended form only) of a circle, parabola and ellipse.
(in picture)

2.Compare the general form of equations of a circle and parabola.
3. Compare the general form of equations of a circle and ellipse.

Compare general equations (expended form only) of a circle, parabola and ellipse. AX2 + Cy2 + Dx + Ey + F = 0 What do they have in common and what is their difference. To complete the task follow the instructions: 2. Compare the general form of equations of a circle and parabola. 3. Compare the general form of equations of a circle and ellipse.

Solution

Solution:

2.

As we know;

parabola: (xh)²=(yk)

1.A parabola with its vertex at ( h, k), opening vertically, will have the following properties.

2.The focus will be at(h,k+1/4a)

3.The directrix will have the equation y=k-1/4a

4.The axis of symmetry will have the equation x = h

and

Circle: (xh)²+(yk)²=r²

where r is the radius of the circle,

and h,k are the coordinates of its center

hence:

r² - (yk)² = (yk)

=> t² + t - r² = 0 (let,y-k =t)

=> t = (1/2)[-1±1 -4r²]

=> y = (1/2)[-1±1 -4r²] + k

intersection of these curves.

3.

As we know;

A circle can be defined as the locus of all points that satisfy the equation

(x-h)² + (y-k)² = r²

where r is the radius of the circle,

and h,k are the coordinates of its center.

whereas:

An ellipse can be defined as the locus of all points that satisfy the equation

x²/a² + y²/b² = 1

where: x,y are the coordinates of any point on the ellipse

and a, b are the radius on the x and y axes respectively

hence on comparing:

1.Distance between the center and any point on the circle is equal, but not in the ellipse

2.The two diameters of an ellipse are different in length, while, in a circle, the size of all the diameters is the same

3.The semi-major axis and semi-minor axis of an ellipse are different in length, while the radius is constant for a given circle