CLASSIFY CONIC SECTION AND WRITE IN STANDARD FORM

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.