CLASSIFY CONIC SECTION AND WRITE IN STANDARD FORM

Standard form of Circle

Equation of circle :

Equation of circle which is having center as (0, 0) and radius r will be in the form.

x2 + y2 = r2

Equation of circle which is having center as (h, k) and radius r will be in the form.

Standard form of Hyperbola

Classify each conic section and write its equation in standard form.

Problem 1 :

25x2 + 9y2 - 36y - 189 = 0

Solution:

25x2 + 9y2 - 36y - 189 = 0

25x2 + 9y2 - 36y = 189

25x2 + 9(y2 - 4y) = 189

25x2 + 9(y2 - 2 • y • 2 + 22 - 22) = 189

25x2 + 9[(y - 2)2 - 4] = 189

By distributing 9, we get

25x2 + 9(y - 2)2 - 36 = 189

25x2 + 9(y - 2)2 = 189 + 36

25x2 + 9(y - 2)2 = 225

Hence, it is Ellipse.

Problem 2 :

-2x2 + 20x + y - 44 = 0

Solution:

-2x2 + 20x + y - 44 = 0

It is parabola. It is symmetric about y-axis.

Problem 3 :

-y2 + 2x + 2y + 3 = 0

Solution:

-y2 + 2x + 2y + 3 = 0

It is parabola. It is symmetric about x-axis.

Problem 4 :

16x2 + 9y2 - 16x + 18y - 131 = 0

Solution:

16x2 + 9y2 - 16x + 18y - 131 = 0

16x2 - 16x + 9y2 + 18y = 131

Now add (1/2)2 and 12 to each side to complete the square on the left side of the equation.

It is Ellipse. Here a2 = 9 and b2 = 16. Since b2 is greater than a2, the ellipse is symmetric about y-axis.

Problem 5 :

x2 + y2 - 8x + 8y + 31 = 0

Solution:

x2 + y2 - 8x + 8y + 31 = 0

It is equation of circle with center (4, -4).

Problem 6 :

2x2 + 2y2 - 14x - 2y + 7 = 0

Solution:

2x2 + 2y2 - 14x - 2y + 7 = 0

It is circle with the center (7/2, 1/2) and radius 3.

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