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Find the equations of the tangent line and the normal line that passes through the point P(0, 4) for the curve:
Solution:
The first thing that we have to do is to check if the given point is included in the curve. Consider the given equation of a curve
The given equation represents a circle because x and y are both in second degree and also have x and y as well. From the coordinates of the given point, substitute x = 0 and y = 4 at the given equation, we have
Since both sides of the equation are equal, then the given point is included in the curve.
Again, consider the given equation of a curve
Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have
The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.
To get the value of the slope of a curve at the given point, substitute x = 0 and y = 4 at the equation above, we have
The slope of a curve at the given point is equal to the slope of tangent line that passes thru also at the given point. Hence,
Therefore, the equation of a tangent line is
Normal line is also a straight line which is perpendicular to tangent line at the point of tangency. Hence,
Therefore, the equation of a normal line is
