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Question 22091: classify the conic section and write each equation in standard form.
x^2-8x+4y+16=0, 3x^2+3y^2-30x+59=0, x^2+2y^2-8x+7=0, 4x^2-y^2+16x+6y-3=0, 3x^2=y^2-4y+3=0, x^2+y^2-2x+1oy+1=0
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! classify the conic section and write each equation in standard form.
YOU HAVE TO WRITE THEM SO THAT ALL SECOND DEGREE TERMS ,TAKING THE FIRST DEGREE TERMS ALSO AS AVAILABLE, ARE PUT IN AS COMBINATION OF PERFECT SQUARES...THEN YOU MAY END UP AS FOLLOWS..IN GENERAL ,THERE ARE FEW OTHER CRITERIA TO BE FOLLOWED ,BUT WE WILL LEAVE THEM FOR THE PRESENT....
1.ALL SECOND DEGREE TERMS COMBINE TO FORM A PERFECT SQUARE..SOME FIRST DEGREE TERMS AND CONSTANT MAY BE LEFT OUT..THEN IT IS A PARABOLA.(PROBLEM 1 BELOW)
2.SECOND DEGREE TERMS OF X COMBINE TO FORM A PERFECT SQUARE AND SECOND DEGREE TERMS OF Y COMBINE TO FORM A PERFECT SQUARE.THEN IT IS AN ELLIPSE OR HYPERBOLA
BASED ON THE FOLLOWING DIFFERENCE
2A.IF THE X AN Y TERMS AS COMBINED INTO PERFECT SQUARES ARE HAVING SAME SIGN THEN IT IS AN ELLIPSE.(PROBLEM 2 BELOW)
2B.IF THE X AN Y TERMS AS COMBINED INTO PERFECT SQUARES ARE HAVING OPPOSITE SIGN THEN IT IS A HYPERBOLA.(PROBLEM 4 BELOW)
1.x^2-8x+4y+16=0
{X^2-2*X*4+4^2}-4^2+4Y+16=0
(X-4)^2+4Y=0
(X-4)^2=4*(-1)Y........THIS IS IN STANDARD FORM
(X-H)^2=4*A*(Y-K) OF PARABOLA...SO THIS IS A PARABOLA
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2.3x^2+3y^2-30x+59=0
{3(X^2-10X)}+3Y^2+59=0
[3{(X^2-2*X*5+5^2)-5^2}]+3Y^2+59=0
[3{(X-5)^2-25}]+3Y^2+59=0
3(X-5)^2-75+3Y^2+59=0
3(X-5)^2+3Y^2-16=0
3(X-5)^2+3Y^2=16.......DIVIDING WITH 16 THROUGH OUT,WE GET..........
(X-5)^2/(16/3)+Y^2/(16/3)=1........THIS IS IN THE STANDARD FORM
(X-H)^2/(A^2)+(Y-K)^2/(B^2)=1.....OF ELLIPSE.SO THIS IS AN ELLIPSE
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3.x^2+2y^2-8x+7=0
TRY YOUR SELF AS ABOVE
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4.4x^2-y^2+16x+6y-3=0
4(X^2+4X)-(Y^2-6Y)-3=0
4{(X^2+2*X*2+2^2)-2^2}-{(Y^2-2*Y*3+3^2)-3^2}-3=0
4(X+2)^2-(Y-3)^2-16+9-3=0
4(x+2)^2-(y-3)^2=10.....DIVIDING THROUGHOUT WITH 10
(X+2)^2/(10/4)-(Y-3)^2/10=1.....THIS IS IN THE STANDARD FORM
(X-H)^2/(A^2)-(Y-K)^2/(B^2)=1....OF HYPERBOLA.SO THIS IS A HYPERBOLA
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5.3x^2=y^2-4y+3=0
TRY YOUR SELF AS ABOVE
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6.x^2+y^2-2x+10y+1=0
TRY YOUR SELF AS ABOVE
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