SOLUTION: Determine the locus of a point P(x,y) so that the product of the slopes joining P(x,y) to (3,-2) and (-2,1) is -6.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Determine the locus of a point P(x,y) so that the product of the slopes joining P(x,y) to (3,-2) and (-2,1) is -6.       Log On


   

Question 1082532: Determine the locus of a point P(x,y) so that the product of the slopes joining P(x,y) to (3,-2) and (-2,1) is -6.

Answer by ikleyn(52814) About Me  (Show Source):
You can put this solution on YOUR website!
.
The literal translation is this equation

%28%28y%2B2%29%2F%28x-3%29%29%2A%28%28y-1%29%2F%28x%2B2%29%29 = -6.


You can transform it:

(y+2)*(y-1) = -6*(x-3)*(x+2),

y%5E2+%2B+y+-2 = -6x%5E2+%2B+6x+%2B+6,

6x%5E2+-+6x+%2B+y%5E2+%2B+y = 8,

6%28x%5E2+-+x%29 + %28y%5E2+%2B+y%29 = 8,

6%2A%28x%5E2+-+2%2A%28x%2F2%29+%2B+%281%2F4%29%29+-+%286%2F4%29%29 + %28y%5E2+%2B+2%2A%28y%2F2%29+%2B+%281%2F4%29%29+-+%281%2F4%29 = 8,

6%28x-%281%2F2%29%29%5E2 + %28y%2B%281%2F2%29%29%5E2 = 8+%2B+6%2F4+%2B+1%2F4,

6%28x-0.5%29%5E2 + %28y%2B0.5%29%5E2 = 9.75,

%28x-0.5%29%5E2%2F%28%289.75%2F6%29%29 + %28y%2B0.5%29%5E2%2F9.75 = 1,

%28x-0.5%29%5E2%2F%28sqrt%289.75%2F6%29%29%5E2%29 + %28y%2B0.5%29%5E2%2F%28%28sqrt%289.75%29%29%5E2%29 = 1,


You got the standard equation of the ellipse. 


Its center is at (0.5,-0.5) .

The major axis is vertical, while the minor axis is horizontal.

The major semi-axis has the length sqrt%289.75%29.

The minor semi-axis has the length sqrt%289.75%2F6%29.

The solution is completed.


Regarding transformations from the general form equation to the standard form and identifying the ellipse elements see the lessons

    - Ellipse definition, canonical equation, characteristic points and elements

    - Standard equation of an ellipse
    - Identify elements of an ellipse given by its standard equation

    - General equation of an ellipse
    - Identify elements of an ellipse given by its general equation
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".