Find the Tangent Line at the Point 2x^2+y^2=54 , (5,2)

Problem

2*x2+y2=54,(5,2)

Solution

1. Differentiate implicitly with respect to x to find the expression for the slope of the tangent line.

(d(2)*x2)/d(x)+d(y2)/d(x)=d(54)/d(x)

4*x+2*yd(y)/d(x)=0

2. Isolate the derivative d(y)/d(x) to solve for the general slope.

2*yd(y)/d(x)=−4*x

d(y)/d(x)=−(4*x)/(2*y)

d(y)/d(x)=−(2*x)/y

3. Substitute the given point (5,2) into the derivative to find the specific slope m

m=−(2*(5))/2

m=−5

4. Apply the point-slope formula y−(y_1)=m*(x−(x_1)) using the point (5,2) and the slope m=−5

y−2=−5*(x−5)

5. Simplify the equation into slope-intercept form y=m*x+b

y−2=−5*x+25

y=−5*x+27

Final Answer

y=−5*x+27

Want more problems? Check here!