document.write( "Question 1081914: The focus of a parabola is (3,−7) and the directrix is y=−4.\r
\n" ); document.write( "\n" ); document.write( " \r
\n" ); document.write( "\n" ); document.write( "What is an equation of the parabola?\r
\n" ); document.write( "\n" ); document.write( " (x−3)2=−6(y+5.5)
\n" ); document.write( " (x−3)2=−3(y+10)
\n" ); document.write( " (x−3)2=−12(y+10)
\n" ); document.write( " (x−3)2=−1.5(y+5.5) \n" ); document.write( "
Algebra.Com's Answer #695966 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
From the definition of parabola, you can find the answer,
\n" ); document.write( "without needing to resort to a memorized formula.
\n" ); document.write( "The definition says that a parabola is the set of all points
\n" ); document.write( "whose distances to vertex and d directrix are equal.
\n" ); document.write( "1) The vertex must be halfway between y=-4 and focus (3,-7),
\n" ); document.write( "so the vertex is (3,-5.5).
\n" ); document.write( "That point satisfies
\n" ); document.write( "\"%28x-3%29%5E2=-6%28y%2B5.5%29\" and \"%28x-3%29%5E2=-1.5%28y%2B5.5%29\" ,
\n" ); document.write( "but not the other two equations.
\n" ); document.write( "2) A point with y=-7 is at a distance
\n" ); document.write( "-4 - (-7) = -4 + 7 = 3 from the directrix,
\n" ); document.write( "so it must be at distance 3 from focus (3,-7).
\n" ); document.write( "That only happens for points (0,-7) and (6,-7).
\n" ); document.write( "Substituting x=0 and y=-7, you find that
\n" ); document.write( "\"%28x-3%29%5E2=%280-3%29%5E2=9\" equals
\n" ); document.write( "\"-6%28y%2B5.5%29=-6%28-7%2B5.5%29=-6%2A%28-1.5%29=9\" ,
\n" ); document.write( "so \"%28x-3%29%5E2=-6%28y%2B5.5%29\" is the equation you are looking for.
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\n" ); document.write( "Your teacher may have told you that the equation of a parabola
\n" ); document.write( "with vertex (h,k), directrix y=k-a and focus (h,k+a) is
\n" ); document.write( "\"%28x-h%29%5E2=4a%28y-k%29\" .
\n" ); document.write( "If so, and if you memorized that formula,
\n" ); document.write( "you easily see that in this case h=3.
\n" ); document.write( "You can find k=-5.5 as the average of
\n" ); document.write( "the -4 in y=-4, and the -7 in (3,-7),
\n" ); document.write( "and you know that k+a=-5.5+a
\n" ); document.write( "is the y-coordinate of the focus,
\n" ); document.write( "so -5.5+a = -7, meaning a= -7 +5.5 = -1.5.
\n" ); document.write( "Applying the memorized formula gives you the same result,
\n" ); document.write( "without needing to understand why you do what you do. \n" ); document.write( "
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