A family of straight lines in the x – y plane is such that each line joins the point (–p, p) on the line y = –x to the point (10 – p, 10 – p) on the line y = x, as shown in Figure 1.38, for different values of p. On a piece of graph paper, draw the lines corresponding to p = 1, 2, 3, · · · , 9. The resulting family is seen to envelop a curve. Show that the line which joins (–p, p) to (10 – p, 10 – p) has equation 5y = 5x – px + 10p – p². Show that two lines of the family pass through the point (x₀, y₀) if x₀² > 20(y₀ – 5), but no lines pass through (x₀, y₀) if x₀²0 – 5). Deduce that the enveloping curve of the family of straight lines is

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(-P.P) Figure 1.38 y = x +5 y (10-p,10-p) X