Step: 1

The foci of the ellipse are (0, - 4) and (0, 4), which are on the y - axis with c = 4.

Step: 2

The center of the ellipse is (0 + 0 2 , - 4 + 4 2 ) = (0, 0).

[Midpoint of the line segment joining foci.]

Step: 3

Length of the semiminor axis is b = 8 2 = 4

Step: 4

[Pythagorean relation.]

Step: 5

So, the standard equation of the ellipse is y 2 3 2 + x 2 1 6 = 1

Correct Answer is : y 2 3 2 + x 2 1 6 = 1

Step: 1

Vertex of the parabola = (0, 0)

Step: 2

Equation of the directrix is y = - 7

Step: 3

Since, y = - 7 is perpendicular to x - axis, x - axis is the focal axis of the parabola.

Step: 4

[Find p by comparing y = - 7 with y = - p .]

Step: 5

So, the equation of the parabola in the standard form is x ^{2} = 4p y or x ^{2} = 4(7)y ⇒ x ^{2} = 28y .

[Substitute p = 7.]

Correct Answer is : x ^{2} = 28y

Step: 1

The foci of the ellipse are (0, - 3) and (0, 3) which are on the y - axis with c = 3.

Step: 2

The center of the ellipse = (0 + 0 2 , - 3 + 3 2 ) = (0, 0).

[Use the mid point formula.]

Step: 3

Length of the semi minor axis is b = 6 2 = 3

Step: 4

[Pythagorean relation.]

Step: 5

So, the standard equation of the ellipse is x 2 9 + y 2 1 8 = 1.

Correct Answer is : x 2 9 + y 2 1 8 = 1

Step: 1

Foci of the ellipse are (6, - 7) and (8, - 7).

Step: 2

Since the y - coordinates of foci are same, the focal axis of the ellipse is parallel to x - axis.

Step: 3

Distance between the foci = 2c = ( 8 - 6 ) 2 + ( - 7 + 7 ) 2 = 2 and hence c = 1

Step: 4

End points of the major axis are (5, - 7) and (9, - 7).

Step: 5

Distance between (5, - 7) and (9, - 7) = 2a = ( 9 - 5 ) 2 + ( - 7 + 7 ) 2 = 4

Step: 6

So, length of the semi major axis = a = 4 2 = 2

Step: 7

[Pythagorean relation.]

Step: 8

Center of the ellipse, (h , k ) = (5 + 9 2 , - 7 - 7 2 ) = (7, - 7)

[Center of the ellipse is the mid point of its major axis.]

Step: 9

So, the equation of the ellipse is ( x - 7 ) 2 4 + ( y + 7 ) 2 3 = 1.

[Use the standard form of the ellipse is ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1.]

Correct Answer is : ( x - 7 ) 2 4 + ( y + 7 ) 2 3 = 1

Step: 1

The equation of a parabola with focus (a , 0) and directrix x = - a is y ^{2} = 4a x .

Step: 2

[Replace a with 6.]

Step: 3

Correct Answer is : y ^{2} = 24x

Step: 1

A parabola with focus (0, c ) and directrix y = - c , equation is given by x ^{2} = 4c y .

Step: 2

Since c = 6 > 0, it open upwards.

Step: 3

[Replace c with 6.]

Step: 4

So, the equation of parabola is x ^{2} = 24y and it open upwards.

Correct Answer is : x ^{2} = 24y , upwards

Step: 1

The equation of a parabola in the standard form whose directrix is the line y = p and focus is the point (0, - p ) is x ^{2} = - 4py .

Step: 2

The directrix is y = 9 and the focus is (0, - 9). So, p = 9.

Step: 3

The equation of the parabola in the standard form is x ^{2} = - 36y .

[Replace p with 9.]

Correct Answer is : x ^{2} = - 36y

Step: 1

A parabola with focus (0, c ) and directrix y = - c , equation is given by x ^{2} = 4cy .

Step: 2

Since c = 3 > 0, it open upwards.

Step: 3

[Replace c with 3]

Step: 4

So, the equation of parabola is x ^{2} = 12y and it open upwards.

Correct Answer is : x ^{2} = 12y , upwards

Step: 1

Foci of the ellipse are (4, - 8) and (10, - 8).

Step: 2

Since the y - coordinates of foci are same, the focal axis of the ellipse is parallel to x - axis.

Step: 3

Distance between the foci = 2c = ( 1 0 - 4 ) 2 + ( - 8 + 8 ) 2 = 6 and hence c = 3.

Step: 4

End points of the major axis are (5, - 8) and (15, - 8).

Step: 5

Distance between (5, - 8) and (15, - 8) = 2a = ( 1 5 - 5 ) 2 + ( - 8 + 8 ) 2 = 10

Step: 6

So, the length of the semimajor axis is a = 10 2 = 5

Step: 7

[Pythagorean relation.]

Step: 8

Since the center of the ellipse is midpoint of its major axis, center of the ellipse is (h , k ) = (5 + 1 5 2 , - 8 - 8 2 ) = (10 , - 8).

Step: 9

So, the equation of the ellipse in the standard form is ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 ⇒ ( x - 1 0 ) 2 2 5 + ( y + 8 ) 2 1 6 = 1.

Correct Answer is : ( x - 1 0 ) 2 2 5 + ( y + 8 ) 2 1 6 = 1

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