# CONVERTING THE GIVEN EQUATION OF CONICS TO THE STANDARD FORM

Equation of circle with center (h, k) :

Equation of parabola with center (h, k) :

Equation of ellipse with center (h, k) :

Equation of hyperbola with center (h, k) :

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)

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