{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW03-solutions - sagbini(ces3287 – HW03 – berg...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: sagbini (ces3287) – HW03 – berg – (55811) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the relation between x and y when the point Q ( x, y ) has the property that dist( Q, P 1 ) + dist( Q, P 2 ) = 4 with respect to points P 1 (- 1 , 0) , P 2 (1 , 0) . 1. x 2 4 + y 2 3 = 1 correct 2. x 2 3 + y 2 4 = 1 3. x 2- y 2 3 = 1 4. y 2- x 2 3 = 1 5. x 2 4- y 2 3 = 1 6. x 2 3 + y 2 = 1 7. x 2 + y 2 3 = 1 8. y 2 3- x 2 4 = 1 Explanation: The distance condition dist( Q, P 1 ) + dist( Q, P 2 ) = 4 ensures that Q traces out an ellipse with foci at P 1 (- 1 , 0) , P 2 (1 , 0) on the x-axis and so has graph P 1 P 2 Q The question really asks for the equation of this ellipse. First notice that dist( Q, P 1 ) = radicalBig ( x + 1) 2 + y 2 , while dist( Q, P 2 ) = radicalBig ( x- 1) 2 + y 2 , so the condition on Q requires that radicalBig ( x + 1) 2 + y 2 + radicalBig ( x- 1) 2 + y 2 = 4 . After squaring both sides and simplifying, this becomes ( x + 1) 2 + y 2 + ( x- 1) 2 + y 2 + 2 radicalBig (( x + 1) 2 + y 2 )(( x- 1) 2 + y 2 ) = 16 . Thus x 2 + y 2 + 1 + radicalBig ( x 2- 1) 2 + 2 y 2 ( x 2 + 1) + y 4 = 8 , which after rearranging and squaring gives 7- ( x 2 + y 2 ) = radicalBig ( x 2 + y 2 ) 2 + 2( y 2- x 2 ) + 1 . Hence, after squaring yet again, we obtain 49- 14( x 2 + y 2 ) = 2( y 2- x 2 ) + 1 . Consequently, the coordinates x, y of Q sat- isfy the equation x 2 4 + y 2 3 = 1 . sagbini (ces3287) – HW03 – berg – (55811) 2 keywords: conic section, ellipse, hyperbola, distance definition, distance, 002 10.0 points Find the relation between x and y when the point Q ( x, y ) has the property that | dist( Q, P 1 )- dist( Q, P 2 ) | = 2 with respect to points P 1 (0 ,- 2) , P 2 (0 , 2) . 1. x 2 3 + y 2 = 1 2. y 2 3- x 2 4 = 1 3. x 2 3 + y 2 4 = 1 4. x 2 4 + y 2 3 = 1 5. x 2 + y 2 3 = 1 6. y 2- x 2 3 = 1 correct 7. x 2- y 2 3 = 1 8. x 2 4- y 2 3 = 1 Explanation: The distance condition | dist( Q, P 1 )- dist( Q, P 2 ) | = 2 ensures that Q traces out a hyperbola with foci at P 1 (0 ,- 2) , P 2 (0 , 2) , on the y-axis and so has graph P 1 P 2 Q The question really asks for the equation of this hyperbola. First notice that dist( Q, P 1 ) = radicalBig x 2 + ( y + 2) 2 , while dist( Q, P 2 ) = radicalBig x 2 + ( y- 2) 2 , so the condition on Q requires that vextendsingle vextendsingle vextendsingle radicalBig x 2 + ( y + 2) 2- radicalBig x 2 + ( y- 2) 2 vextendsingle vextendsingle vextendsingle...
View Full Document

{[ snackBarMessage ]}

Page1 / 8

HW03-solutions - sagbini(ces3287 – HW03 – berg...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online