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11.5. Find the locus of points in rhombus ABCD in Fig. 11-16, which are equidistant from (a) AB and AD; (b) AB and (11.1) BC; (c) A and C; (d) B and D; (e) each of the four sides. 11.6. In Fig. 11-17, find the locus of points which are on or inside circle C and (a) 5 units from O (b) 15 units from O (c) Equidistant from circles A and C (d) 10 units from circle C (e) 10 units from circle A (f) 5 units from circle B (g) The center of a circle tangent to circles A and C (11.1 and 11.2)

on the other hand, is not memoryless because the output at time no depends on the value of the input both at time no and at time no - 1.

An additive system is one for which the response to a sum of inputs is equal to the sum of the inputs individually. Thus,

Fig. 11-17

(k) The region bounded by the parabola x = y2 + 2 and the line y = x 8. (l) The region bounded by the parabolas y = x 2 x and y = x x 2 .

for any signals X I (n) and x2(n).

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11.7. Determine the locus of the center of (a) a coin rolling around and touching a smaller coin; (b) a coin rolling around and touching a larger coin; (c) a wheel moving between two parallel bars and touching both of them; (d) a wheel moving along a straight metal bar and touching it. (11.2) 11.8. Find the locus of points that are in rectangle ABCD of Fig. 11-18 and the center of a circle (a) Tangent to AD and BC (b) Tangent to AB and CD (c) Tangent to AD and EF (d) Of radius 10, tangent to BC (e) Of radius 20, tangent to AD (f) Tangent to BC at G (11.2)

A system is said to be homogeneous if scaling the input by a constant results in a scaling of the output by the same amount. Specifically,

Fig. 11-18

T [cx(n)]= cT [x(n)]

11.9. Locate each of the following:

for any complex constant c and for any input sequence x(n).

(m) The region in the rst quadrant bounded by the curves y = x 2 and y = x 4 . (n) The region between the curve y = x 3 and the lines y = x and y = 1.

~ ( n= )

(11.4)

x(n - 1 )

(a) Treasure that is buried 5 ft from a straight fence and equidistant from two given points where the fence meets the ground (b) Points that are 3 ft from a circle whose radius is 2 ft and are equidistant from two lines which are parallel to each other and tangent to the circle (c) A point equidistant from the three vertices of a given triangle (d) A point equidistant from two given points and equidistant from two given parallels (e) Points equidistant from two given intersecting lines and 5 ft from their intersection (f) A point that is equidistant from the sides of an angle and 1 in from their intersection 2 11.10. Locate the point or points which satisfy the following conditions with respect to ^ABC in Fig. 11-19: (a) Equidistant from its sides (b) Equidistant from its vertices (c) Equidistant from A and B and from AB and BC (d) Equidistant from BC and AC and 5 units from C (e) 5 units from B and 10 units from A (11.4)

x2(n)

Fig. 11-19

32.9 Find the lengths of the following curves: (a) y = 1 x4 + 2 from x = 1 to x = 2. 8 4x (b) y = 3x 2 from x = 0 to x = 1. (c) y = x 2/3 from x = 1 to x = 8. (d) x 2/3 + y2/3 = 4 from x = 1 to x = 8. (e) y = 1 x5 + from x = 1 to x = 2. 15 4x 3 1 x(3 x) from x = 0 to x = 3. (f) y = 3 2 (1 + x 2 )3/2 from x = 0 to x = 3. 3

y(n) =x(n)

Fig. 11-20

+x'(n - 1 )

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