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The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 ≤ 5. If point (- 5, 0) is the edge of the shadow, how far above the x-axis is the lamp located?
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| Photo by Math Principles in Everyday Life |
Solution:
As you can see from the figure that the two lines which are the edges of the shadow are tangent to the elliptical region. In this problem, we will use only one line which contains a point (- 5, 0). Let's consider the given equation of the elliptical region
Take the derivative on both sides of the equation with respect to y to get the slope of a curve by implicit differentiation
You cannot substitute (-5, 0) to the above equation because the point (-5, 0) is not in the curve. Let (x, y) be the point of intersection of a line and an ellipse. In short, its tangent point. Get the equation of a tangent line by point-slope form, we have
Substitute (-5, 0) to the above equation
Equate the slope of a line with the slope of a curve
Substitute the above equation to the equation of an ellipse
Solve for the value of y
Since their point of intersection or tangent point is located in second quadrant, choose the positive value which is y = 1. The tangent point is (- 1, 1). The slope of a curve at (- 1, 1) is
The slope of a curve is the same as the slope of a line because the line is tangent to the curve. We can get the equation of a tangent line using the point-slope form
Substitute (- 5, 0) and m = ¼ to the above equation
From the word problem, the lamp is located 3 units from the y-axis which is x = 3. We can solve the height of the lamp from the ground or x-axis by substitute x = 3 to the above equation
Therefore, the height of the lamp from the ground or x-axis is 2 units.