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Ex. 1508.

centre.

Ex. 1509. height 3 cm.

Repeat Ex. 1506 with a chord that subtends 90° at the

Find the area of a segment whose chord is 12 cm. and Also find the ratio of the segment to the rectangle of the same base and height.

Ex. 1510. Find the area of a segment of base 10 cm. and height 5 cm.

Ex. 1511. Find the area of a segment of base 4 cm. and height 8 cm.

Ex. 1512.

A square is inscribed in a circle of radius 2 in. Find the area of a segment cut off by a side of the square.

Ex. 1513. From a point outside a circle of radius 10 cm., a pair of tangents are drawn to the circle; the angle between the tangents is 120°. Find the area included between the two tangents and the circumference.

SECTION XII.

Ex. 1514.

FURTHER EXAMPLES OF LOCI.

Plot the locus of points the sum of whose distances from two fixed points remains constant.

(Mark two points S, H, say, 4 in. apart. Suppose that the point P moves so that SP+HP=5 in. Then the following are among the possible pairs of values:

SP 4.5 4.0 3.5

3.0 2.5

2.0 1.5 1.0 0.5

HP 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Plot all the points corresponding to all these distances, by means of intersecting arcs. Why were not values such as SP=47, HP=0.3 included in the above table? Draw a neat curve, free-hand, through all these points. The locus is an oval curve called an ellipse.)

Ex. 1515. What kinds of symmetry are possessed by an ellipse?

Ex. 1516. Describe an ellipse mechanically as follows. Stick two pins into the paper about 4 in. apart; make a loop of fine string, gut or cotton and place it round the pins (see fig. 288).. Keep the loop extended by means of the point of a pencil, and move the point round the pins. It will, of course, describe an ellipse.

fig. 288.

Ex. 1517. Plot the locus of points the difference of whose distances from two fixed points remains constant.

(For example, let the two fixed points S, H be 4 in. apart, and let the constant difference be 2 in. Make a table as in Ex. 1514. Remember to make SP HP for some points, HP>SP for other points.)

This curve is called a hyperbola.

Ex. 1518. Plot the locus of points the product of whose distances from two fixed points remains constant.

(For example, mark two points S, H exactly 4 in. apart. First, to plot the locus SP. HP=5.

Fill up the blanks in the following table:

SP 5 4.8 4 3 No5 2

HP

3 4 4.8 5

Secondly, plot the locus SP. HP=4; thirdly, plot the locus SP. HP=3. All three loci should be drawn in the same figure.

The first locus will be found to resemble a dumb-bell, the second a figure of 8; the third consists of two separate ovals.)

Ex. 1519. Plot the locus of a point which moves so that the ratio of its distances from two fixed points remains constant.

SP

HP

(For example, let the two fixed points S, H be taken 3 in. apart; and let

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Ex. 1520. OP is a variable chord passing through a fixed point O on a circle; OP is produced to Q so that PQ=OP; find the locus of Q.

Ex. 1521. A point moves so that its distance from a fixed point S is always equal to its distance from a fixed line MN: find its locus. (This is best done on inch paper. Take the point S 2 in. distant from the line MN. Then plot points as follows. What is the locus of points distant 3 in. from MN? distant 3 in. from S? The intersection of these two loci gives two positions of the required point. Similarly find other points.)

x2

The curve obtained is called a parabola. It is the same curve as would be obtained by plotting the graph y= +1, taking for axis of x the line MN, and for avis of y the perpendicular from S to MN. It is remarkable as being the curve described by a projectile, e.g. a stone or a cricket-ball. Certain comets move in parabolic orbits, the sun being situated at the point S.

Ex. 1522. A point moves in a plane subject to the condition that its distance from a fixed point S is always in a fixed ratio to its distance from a fixed straight line MN. Plot the curve described.

(i) Let the distance from S be always half the distance from MN. Take S 3 in. from MN.

(ii) Let the distance from S be always twice the distance from MN. Take S 3 in. from MN.

These curves will be recognized as having been obtained already.

Ex. 1523. Plot the locus of a point on the connecting-rod of a steamengine.

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(The upper diagram in fig. 289 represents the cylinder, piston-rod (AB), connecting-rod (BC), and crank (CD) of a locomotive. In the lower diagram the different parts are reduced to lines. B moves to and fro along a straight line, C moves round a circle. Take BC 10 cm., CD=3 cm. Plot the locus of a point P on BC, where BP is (i) 1cm., (ii) 5 cm., (iii) 9 cm. This may be done, either by drawing a large number of different positions of BC; or, much more easily, by means of tracing paper. Draw BD and the circle on your drawing paper, BC on tracing paper. Keep the two ends of BC on the straight line and circle respectively, and prick through the different positions of P.)

Ex. 1524. A rod moves so that it always passes through a fixed point while one end always lies on a fixed circle. Plot the locus of the other end.

(Tracing paper should be used. A great variety of curves may be obtained by varying the position of point and circle, and the length of the rod. It will be seen that this exercise applies to the locus of a point on the piston-rod of an oscillating cylinder; also to the locus of a point on the stay-bar of a casement window.)

Ex. 1525. The ends of a rod slide on two wires which cross at right angles. Find the locus of a point on the rod.

(Represent the rod by a line of 10 cm.; take the point 3 cm. from one end of the rod; also plot the locus of the mid-point. Use tracing paper.)

Ex. 1526. Two points A, B of a straight line move along two intersecting lines at right angles. Plot the locus of a point P, in AB produced. [Tracing paper.]

fig. 290.

Ex. 1527. Draw two intersecting lines. On tracing paper mark three points A, B, C. Make A slide along one line and B along the other; plot the locus of C. Ex. 1528. Draw two equal circles of radius 4 cm., their centres being 10 cm. apart. The two ends of a line PQ, 10 cm. in length, slide one along each circle. Plot the locus of the mid-point of PQ; also of a point 1cm. from P.

(Most quickly done with tracing paper. It is easy to construct a model machine to describe the curve.)

Ex. 1529. Draw two circles. On tracing paper mark three points A, B, C. Make A slide along one circle, B along another, and plot the locus of C. (Experiment with different circles and arrangements of points. You will find that in at least one case the locus-curve shrinks to a single point.)

Ex. 1530. OA, AP are two rods jointed at A. OA revolves about a hinge at O, and AP revolves twice as fast as OA, in the same direction. Find the path of a point on AP. (Make OA=2 in., AP=211⁄2 in. Plot the locus of

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P; also of Q and R, taking AQ=2 in., AR=11⁄2 in. To draw the different positions of the rod, notice that when OA has turned through, say, 30°, AP has turned through 60° and therefore makes an angle of 30° with OA produced.)

The loci are different forms of the limaçon; the locus of Q is heartshaped, and is called a cardioid. The locus of P has a small loop in it.

Ex. 1531. Repeat Ex. 1530, with the difference that, as OA revolves, AP remains parallel to its original position.

Ex. 1532. Two equal rods OA, AQ, jointed as in Ex. 1530, revolve in opposite directions at the same rate. Find the locus of Q and of the midpoint of AQ.

Ex. 1533.

O is a fixed point on a circle of radius 1 in. OP, a variable chord, is produced to Q, PQ being a fixed length; also PQ' (=PQ) is marked off along PO. Plot the locus of Q and Q' when PQ is (i) 2 in., (ii) 2 in., (iii) 11⁄2 in.

(Draw a long line on tracing paper, and on it mark P, Q and Q'.)

Ex. 1534. Through a fixed point S is drawn a variable line SP to meet a fixed line MN in P. From P a fixed length PQ is measured off along SP (or SP produced). Find the locus of Q.

(Use tracing paper.

Take S 1 in. from MN. Plot the locus of Q
(i)_ when PQ=1 in., measured from P away from S,
(ii) when PQ=1 in., measured from P towards S,
(iii) when PQ=2 in., measured from P towards S.)
The curves obtained are different forms of the conchoid.

Ex. 1535. A company of soldiers are extended in a straight line. At a given signal, they all begin to move towards a certain definite point, at the regulation pace. Are they in a straight line after 3 minutes? If not, what curve do they form?

Ex. 1536. XOX', YOY' are two fixed straight lines, C is a fixed point (see fig. 292). A variable line PQ is drawn through C to meet XOX', YOY' in P, Q respectively. Plot the locus of the midpoint of PQ.

(Let XOX', YOY' intersect at 60°, and take C on the bisector of LXOY, 5 cm. from O.)

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fig. 292.

P, X

Ex. 1537. (Inch paper.) Draw a circle of radius 2 in. and a straight line distant 6 in. from the centre of the circle. P is a variable point on the circle; Q is the mid-point of PN, the perpendicular from P upon line. Plot the locus of Q.

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