Parabola Circle

The parabola is a conic section to which the locus of points which moves in that plane is always equidistant from a fixed point called the focus.

A circle is a simple shape of geometry consisting of those points in a plane which is equidistant from a given point called the center. The distance of the points of circle from its center is called its radius(Source: wikipedia).

Understanding Definition of Parabola is always challenging for me but thanks to all math help websites to help me out.

Standard form of Parabola and Circle:

Parabola:

The standard form of parabola is y2 = 4ax

Where ‘a’ is the focal distance of the parabola.

Circle:

The circle equation with centre (0, 0) and radius r is

x2 + y2 = r2

The circle equation with centre (h, k) and radius r is

(x – h)2 + (y – k)2 = r2

Problems on parabola and Circle:

Prob 1:

Find the equation of the circle with radius 5 and centre (3, -2).

Sol:

Given centre (h, k) = (3, -2)

Radius r = 5

The formula for the equation of circle = (x – h)2 + (y – k)2 = r2

(x – 3) 2 + (y – (-2)) 2b = 52

(x – 3) 2 + (y + 2)) 2 = 52

(x – 3) 2 + (y + 2)) 2 = 25

Prob 2:

Find the radius and centre of the circle x2 + y2 + 8x + 6y = 0

Sol:

We complete the square on the x-related part and on the y-related part, at the same time.

x2 + y2 + 8x + 6y = 0

Group the x parts together and the y parts together:

x2 + 8x + y2 + 6y = 0

Complete the square on each of the x and y parts.

x2 + 8x + 16 + y2 + 6y + 9 =25

(x + 4)2 (y + 3)2 = 52

Hence centre = (-4, -3)

radius = 5

Prob 3:

Find the equation of the parabola with focus (4,0) and directrix x = 5.

Sol:

Given focus (4, 0)

Directrix = 5

The equation of parabola is given by

y2 = 4ax

= 4 (5) x

y2  = 20x

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Prob 4:

Find the equation of the parabola having vertex at origin along the x-axis and passing through (3,-2).

Sol:

The general formula is y2 = 4ax

Given (x, y) = (3, -2)

Therefore, y2 = 4ax

(-2)2 = 4a (3)

4 = 12a

a = 1/3

Therefore  the required equation is y2 = 4/3 x