A geometry problem by César Emanuel Castro
Identify the conic section 9 x 2 + 4 y 2 − 1 8 x + 1 6 y − 1 1 = 0 .
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2 solutions
The above equation can be rewritten as:
( 9 x 2 − 1 8 x + 9 − 9 ) + ( 4 y 2 + 1 6 y + 1 6 − 1 6 ) − 1 1 = 0 ;
or 9 ( x − 1 ) 2 + 4 ( y + 2 ) 2 = 3 6 ;
or 4 ( x − 1 ) 2 + 9 ( y + 2 ) 2 = 1 ⇒ E l l i p s e .
Since
(i) the quadratic terms don't make a perfect square (chance of a parabola is eliminated) ,
(ii) there is no term involving x y (chance of a pair of straight lines is eliminated) ,
(iii) there is a + ve sign between term involving x 2 and the term involving y 2 (chance of a hyperbola is eliminated) , and
(iv) the coefficients of x 2 and y 2 are not equal (chance of a circle is eliminated) ,
therefore the given equation represents an ellipse .