#### Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity   Option 1) 7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0 Option 2) 7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0 Option 3) 7x2 - 2y2 + xy - 14x + 2y - 22 = 0 Option 4) none of the above

Eccentricity -

The ratio of a distance of point from focus to distance from fixed line.

- wherein

It is donated by .

&

Directrix -

The fixed straight line of a conic section.

- wherein

&

Focus -

The fixed point of a conic section

- wherein

Option 1)

7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0

This solution is correct

Option 2)

7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0

This solution is incorrect

Option 3)

7x2 - 2y2 + xy - 14x + 2y - 22 = 0

This solution is incorrect

Option 4)

none of the above

This solution is incorrect
The text passage contains a mathematical question about finding the equation of a hyperbola given its directrix, focus, and eccentricity. It also provides multiple-choice options and indicates which one is correct.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
#### Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity   Option 1) 7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0 Option 2) 7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0 Option 3) 7x2 - 2y2 + xy - 14x + 2y - 22 = 0 Option 4) none of the above

Eccentricity -

The ratio of a distance of point from focus to distance from fixed line.

- wherein

It is donated by .

&

Directrix -

The fixed straight line of a conic section.

- wherein

&

Focus -

The fixed point of a conic section

- wherein

Option 1)

7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0

This solution is correct

Option 2)

7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0

This solution is incorrect

Option 3)

7x2 - 2y2 + xy - 14x + 2y - 22 = 0

This solution is incorrect

Option 4)

none of the above

This solution is incorrect
user
The text passage contains a mathematical question about finding the equation of a hyperbola given its directrix, focus, and eccentricity. It also provides multiple-choice options and indicates which one is correct.

TYPE: SINGLE_QUESTION_WITH_ANSWER