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已知x=8t/4+t2,y=16-4t2/4+t2求证x2/4+y2/16=1
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已知x=8t/4+t2,y=16-4t2/4+t2 求证x2/4+y2/16=1
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答案和解析
x²=[8t/4+t²]²=64t²/(4+t²)²
y²=[(16-4t²)/(4+t²)]²=[16²-128t²+16t^4]/(4+t²)²
所以 x²/4+y²/16=16t²/(4+t²)²+[16-8t²+t^4]/(4+t²)²
=(16+8t²+t^4)/(4+t²)²
=(4+t²)²/(4+t²)²
=1
所以 x²/4+y²/16=1
y²=[(16-4t²)/(4+t²)]²=[16²-128t²+16t^4]/(4+t²)²
所以 x²/4+y²/16=16t²/(4+t²)²+[16-8t²+t^4]/(4+t²)²
=(16+8t²+t^4)/(4+t²)²
=(4+t²)²/(4+t²)²
=1
所以 x²/4+y²/16=1
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