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ZanzabumX [31]
3 years ago
12

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 7x2 + xy + 7y2 = 15, (1, 1

) (ellipse)
Mathematics
1 answer:
riadik2000 [5.3K]3 years ago
7 0
Assume y=y(x).

7x^2+xy+7y^2=15
\dfrac{\mathrm d}{\mathrm dx}[7x^2+xy+7y^2]=\dfrac{\mathrm d}{\mathrm dx}[15]
14x+y+x\dfrac{\mathrm dy}{\mathrm dx}+14y\dfrac{\mathrm dy}{\mathrm dx}=0
\dfrac{\mathrm dy}{\mathrm dx}(x+14y)=-(14x+y)
\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac{14x+y}{x+14y}

The slope of the tangent line to the curve at (a,b) is then -\dfrac{14a+b}{a+14b}, so the tangent to (1,1) has slope

\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=1,y=1}=-\dfrac{15}{15}=-1

The point-slope form of the tangent line is then

y-1=-(x-1)\iff y=-x+2
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