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Find the shortest distance from the point P(2, 0) to a point on the curve y2 - x2 = 1, and find the point on the curve closest to P.
Solution:
To illustrate the problem, it is better to draw the figure and sketch the graph of the curve (hyperbola) as well as the given point in Rectangular Coordinate System as follows
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| Photo by Math Principles in Everyday Life |
As you can see from the figure above that there will be two distances of a point to a curve because the curve is hyperbola and it is symmetrical with x axis and y axis. The other point on the curve is a point of tangency. The distance of a point to the point of tangency is the perpendicular distance of a point to the tangent line. To understand more the problem, label further the above figure as follows
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| Photo by Math Principles in Everyday Life |
The distance of two points is
If the first point is P(2, 0), then the above equation becomes
If the second point is on the curve, then the above equation becomes
Next, we need to eliminate y at the above equation. If the equation of a curve is
,then the value of y will be equal to
Substitute the value of y to the first equation which is the distance of two points, we have
Take the derivative of the above equation with respect to x, we have
Set dd/dx = 0 because we want to minimize the distance of a point to a curve.
Substitute the value of x to the equation of the curve in order to get the value of y, we have
Therefore, the point on the curve is
The shortest distance of a point to a curve is
