To find the equation of the tangent line, we first need to find the derivative of the given curve. The equation of the curve is given by:
x^2y + y^3x + 3x - 13 = 0
Taking the derivative of both sides with respect to x:
2xy + x^2(dy/dx) + 3 + 3y^2x(dy/dx) = 0
Rearranging the terms:
(dy/dx)(x^2 + 3y^2x) = -2xy - 3
(dy/dx) = (-2xy - 3)/(x^2 + 3y^2x)
Now, to find the slope of the tangent line at the point (1,2), substitute x = 1 and y = 2 into the derivative:
(dy/dx) = (-2(1)(2) - 3)/(1^2 + 3(2)^2(1))
(dy/dx) = (-4 - 3)/(1 + 12)
(dy/dx) = -7/13
So the slope of the tangent line is -7/13.
To find the equation of the tangent line at the point (1,2), we can use the point-slope form:
y - y1 = m(x - x1)
where (x1, y1) = (1, 2) and m = -7/13:
y - 2 = (-7/13)(x - 1)
Multiply both sides by 13 to eliminate the fraction:
13y - 26 = -7(x - 1)
13y - 26 = -7x + 7
Rearrange the equation to get it in the standard form:
7x + 13y = 33
So the equation of the tangent line to the curve at the point (1,2) is 7x + 13y = 33.
To find the equation of the normal line, we need to find the negative reciprocal of the slope of the tangent line. The negative reciprocal of -7/13 is 13/7.
Using the same point-slope form as before, with m = 13/7:
y - 2 = (13/7)(x - 1)
Multiply both sides by 7 to eliminate the fraction:
7y - 14 = 13(x - 1)
7y - 14 = 13x - 13
Rearrange the equation to get it in the standard form:
13x - 7y = 1
So the equation of the normal line to the curve at the point (1,2) is 13x - 7y = 1.
Find the equation of the tangent and the normal lines to the curves x^2y+y^3x+3x-13=0 at the point (1,2)
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