Explanation
To find the equation of the ellipse, we need to determine the values of a2 and b2 for the standard form a2x2+b2y2=1.
1. Identify Given Parameters
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Eccentricity (e): 31
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Directrix: x=9
Since the directrix is of the form x=d (a vertical line), the major axis of the ellipse lies along the x-axis.
2. Find the Semi-Major Axis (a)
The formula for the directrix of an ellipse centered at the origin is x=±ea.
Given x=9:
Substitute the value of e=31:
Thus, a2=32=9.
3. Find the Semi-Minor Axis (b)
We use the eccentricity relation b2=a2(1−e2):
4. Formulate the Equation
Substitute a2=9 and b2=8 into the standard equation a2x2+b2y2=1:
To match the options, multiply the entire equation by the LCM of 9 and 8 (which is 72):
Final Answer:
The equation of the ellipse is 8x2+9y2=72. The correct option is B.
