Answer to Question #114931 in Analytic Geometry for ANJU JAYACHANDRAN

Question #114931
What is the new equation of the conic
x^2 +y^2 +4x−2y+3 = 0, when
i) the origin is shifted at (2,−1), followed by a rotation of axes through 45^◦
?
ii) the axes are rotated through 45^◦
, followed by the shifting of the origin at
(2,−1)?
Are the equations in i) and ii) above the same? Why?
1
Expert's answer
2020-05-11T18:21:45-0400

Solution.

x2+y2+4x2y+3=0;x^2+y^2+4x-2y+3=0;

i)the origin is shifted at (2,−1)

x=x+2;y=y1;x'=x+2; y'=y-1;

(x+2)2+(y1)2+4(x+2)2(y1)+3=0;(x+2)^2+(y-1)^2+4(x+2)-2(y-1)+3=0;

x2+4x+4+y22y+1+4x+82y+2+3=0;x^2+4x+4+y^2-2y+1+4x+8-2y+2+3=0;

x2+y2+8x4y+18=0;x^2+y^2+8x-4y+18=0;

the axes are rotated through 45o;

x=xcosαysinα;y=xsinα+ycosα;x'=xcos\alpha-ysin\alpha; y'=xsin\alpha+ycos\alpha;

x=x22y22=22(xy);x'=x\dfrac{\sqrt{2}}{2}-y\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{2}}{2}(x-y);

y=x22+y22=22(x+y);y'=x\dfrac{\sqrt{2}}{2}+y\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{2}}{2}(x+y);

12(xy)2+12(x+y)2+42(xy)22(x+y)+18=0;\dfrac{1}{2}(x-y)^2+\dfrac{1}{2}(x+y)^2+4\sqrt{2}(x-y)-2\sqrt{2}(x+y)+18=0;

12x2xy+12y2+12x2+xy+12y2+42x42y22x22y+18=0;\dfrac{1}{2}x^2-xy+\dfrac{1}{2}y^2+\dfrac{1}{2}x^2+xy+\dfrac{1}{2}y^2+4\sqrt{2}x-4\sqrt{2}y- 2\sqrt{2}x-2\sqrt{2}y+18=0;

x2+y2+22x62y+18=0;x^2+y^2+2\sqrt{2}x-6\sqrt{2}y+18=0;

ii)the axes are rotated through 45o;

x=xcosαysinα;y=xsinα+ycosα;x'=xcos\alpha-ysin\alpha; y'=xsin\alpha+ycos\alpha;

x=x22y22=22(xy);x'=x\dfrac{\sqrt{2}}{2}-y\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{2}}{2}(x-y);

y=x22+y22=22(x+y);y'=x\dfrac{\sqrt{2}}{2}+y\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{2}}{2}(x+y);

12(xy)2+12(x+y)2+22(xy)2(x+y)+3=0;\dfrac{1}{2}(x-y)^2+\dfrac{1}{2}(x+y)^2+2\sqrt{2}(x-y)-\sqrt{2}(x+y)+3=0;

12x2xy+12y2+12x2+xy+12y2+22x22y2x2y+3=0;\dfrac{1}{2}x^2-xy+\dfrac{1}{2}y^2+\dfrac{1}{2}x^2+xy+\dfrac{1}{2}y^2+2\sqrt{2}x-2\sqrt{2}y- \sqrt{2}x-\sqrt{2}y+3=0;

x2+y2+2x32y+3=0;x^2+y^2+\sqrt{2}x-3\sqrt{2}y+3=0;

the origin is shifted at (2,−1)

x=x+2;y=y1;x'=x+2; y'=y-1;

(x+2)2+(y1)2+2(x+2)32(y1)+3=0;(x+2)^2+(y-1)^2+\sqrt{2}(x+2)-3\sqrt{2}(y-1)+3=0;

x2+4x+4+y22y+1+2x+2232y+32+3=0;x^2+4x+4+y^2-2y+1+\sqrt{2}x+2\sqrt{2}-3\sqrt{2}y+3\sqrt{2}+3=0;

x2+y2+x(4+2)y(2+32)+8+52=0;x^2+y^2+x(4+\sqrt{2})-y(2+3\sqrt{2})+8+5\sqrt{2}=0;


Answer: i)x2+y2+22x62y+18=0;x^2+y^2+2\sqrt{2}x-6\sqrt{2}y+18=0;

ii)x2+y2+x(4+2)y(2+32)+8+52=0;x^2+y^2+x(4+\sqrt{2})-y(2+3\sqrt{2})+8+5\sqrt{2}=0;

the equations i) and ii) are different.


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