Answer :
The general equation of an ellipse with its major axis parallel to the x-axis and center at (h, k) is:
\[
\frac{{(x - h)^2}}{{a^2}} + \frac{{(y - k)^2}}{{b^2}} = 1
\]
where:
- (h, k) is the center of the ellipse,
- a is the semi-major axis length (half the length of the major axis),
- b is the semi-minor axis length (half the length of the minor axis).
Given that the major axis lengths are twice the minor axis lengths, we have \(a = 2b\).
Now, let's compare the given equations with the standard form of the ellipse equation:
1. \(4x^2+25y^2+32x-250y+589=0\)
- Not in standard form.
2. \(4x^2+y^2+16x+4y+4=0\)
- Not in standard form.
3. \(16x^2+y^2-64x+8y+16=0\)
- Not in standard form.
4. \(2x^2+8y^2-12x+16y-174=0\)
- Not in standard form.
5. \(3x^2+12y^2+18x-24y-69=0\)
- Not in standard form.
6. \(x^2+9y^2-2x+18y-71=0\)
- Not in standard form.
Since none of the given equations are in standard form, we cannot directly identify the equations of ellipses with major axis lengths twice their minor axis lengths from the given options.
\[
\frac{{(x - h)^2}}{{a^2}} + \frac{{(y - k)^2}}{{b^2}} = 1
\]
where:
- (h, k) is the center of the ellipse,
- a is the semi-major axis length (half the length of the major axis),
- b is the semi-minor axis length (half the length of the minor axis).
Given that the major axis lengths are twice the minor axis lengths, we have \(a = 2b\).
Now, let's compare the given equations with the standard form of the ellipse equation:
1. \(4x^2+25y^2+32x-250y+589=0\)
- Not in standard form.
2. \(4x^2+y^2+16x+4y+4=0\)
- Not in standard form.
3. \(16x^2+y^2-64x+8y+16=0\)
- Not in standard form.
4. \(2x^2+8y^2-12x+16y-174=0\)
- Not in standard form.
5. \(3x^2+12y^2+18x-24y-69=0\)
- Not in standard form.
6. \(x^2+9y^2-2x+18y-71=0\)
- Not in standard form.
Since none of the given equations are in standard form, we cannot directly identify the equations of ellipses with major axis lengths twice their minor axis lengths from the given options.
