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Finding a Distance and the Midpoint of a Line Segment In Exercises 3-12, (a) plot the poins, (b) find the distance between the points and (c) find the midpoint of the line segment joining the points.

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Patrick Cosico
Patrick CosicoLv9
2 Nov 2020

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FInd the coordinate of the midpoint of the line joining 4 and -4; 3 and -5; -1.7 and 3.7; sqrt 2 and sqrt 3; and x1 and x2.

Question:

Find the coordinates of the midpoint of the line joining the points

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b.(1.7,3.7) and ( , )

A tangent line is drawn to the hyperbola xy − c at a point P.

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(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter

Where P is located on the hyperbola.

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on, so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.

(a) Find an expression for the intensity at the point P.

(b) If m, use graphs of and to show that the intensity is minimized when m, that is, when P is at the midpoint of ,.

(c) If m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint.

(d) Somewhere between m and m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d.

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