CONVERTING THE GIVEN EQUATION OF CONICS TO THE STANDARD FORM

Convert the equation to standard form by completing the square. Then identify what type of conic section the equation represents.

If it is a circle, ellipse, or hyperbola, then name its center. If it is a parabola, then name its vertex.

Problem 1 :

x² + 6x + 8y + 1 = 0

Solution:

x² + 6x + 8y + 1 = 0

(x² + 6x) + 8y = -1

(x² + 6x + 9) + 8y = -1 + 9

(x + 3)² + 8y = 8

(x + 3)² = -8(y - 1)

This is a parabola.

Vertex (h, k) = (-3, 1)

Problem 2 :

9y² - 4x² - 18y + 24x - 63 = 0

Solution:

9y² - 4x² - 18y + 24x - 63 = 0

9y² - 18y - 4x² + 24x = 63

9(y² - 2y + 1) - 4(x² - 6x + 9) = 63 + 9 - 36

9(y - 1)² - 4(x - 3)² = 36

This is a hyperbola.

Here (h, k) is a center of hyperbola.

C = (3, 1)

Problem 3 :

4x² + 36y - 32x + 9y² + 64 = 0

Solution:

4x² + 36y - 32x + 9y² + 64 = 0

4x² - 32x + 9y² + 36y = -64

4(x² - 8x + 16) + 9(y² + 4y + 4) = -64 +  64 + 36

4(x - 4)² + 9(y + 2)² = 36

This is an ellipse.

Here (h, k) is a center of ellipse.

C = (4, -2)

Problem 4 :

9x² + 16y² - 18x + 64y - 71 = 0

Solution:

9x² + 16y² - 18x + 64y - 71 = 0

9x² - 18x + 16y² + 64y = 71

9(x² - 2x + 1) + 16(y² + 4y + 4) = 71 + 9 + 64 

9(x - 1)² + 16(y + 2)² = 144

This is an ellipse.

Here (h, k) is a center of ellipse.

C = (1, -2)

Problem 5 :

4x² - y² + 32x + 6y + 39 = 0

Solution:

4x² - y² + 32x + 6y + 39 = 0

4x² + 32x - y² + 6y = -39

4(x² + 8x + 16) - (y² - 6y + 9) = -39 + 64 - 9

4(x + 4)² - (y - 3)² = 16

This is a hyperbola.

Here (h, k) is a center of hyperbola.

C = (-4, 3)

Problem 6 :

x² + y² - 6x + 8y = 1

Solution:

x² + y² - 6x + 8y = 1

x² - 6x + y² + 8y = 1

(x² - 6x + 9) + (y² + 8y + 16) = 1 + 9 + 16

(x - 3)² + (y + 4)² = 26

This is a circle.

Center (h, k) = (3, -4)

Problem 7 :

4x² + 4y² - 24x + 32y - 4 = 0

Solution:

4x² + 4y² - 24x + 32y - 4 = 0

4x² - 24x + 4y² + 32y = 4

4(x² - 6x + 9) + 4(y² + 8y + 16) = 4 + 36 + 64

4(x - 3)² + 4(y + 4)² = 104

(x - 3)² + (y + 4)² = 104/2

(x - 3)² + (y + 4)² = 26

This is a circle.

Center (h, k) = (3, -4)

Problem 8 :

y² + 8y - 4x + 8 = 0

Solution:

y² + 8y - 4x + 8 = 0

(y² + 8y + 16) - 4x = -8 + 16

(y + 4)² - 4x = 8

(y + 4)² = 4x + 8

(y + 4)² = 4(x + 2)

This is a parabola.

Vertex (h, k) = (-2, -4)