1969-04-15 — Page 23

育教僑華真三第張六第日九十月二年西己舾夏WAH - KIU YAT PO

報日僑華

二期星日五十月四年九六九一曆公年八十五國民華中

1969FU

數學科

(廿四)

歐陽鐇文 ̇

MATHEMATICS (21)

Plies on st. line. BF

ie. the locus of P is a st. line through

the locus of a is bu

HINTS & ANS. TO EI. 23

2) PROOF: Join AB

ZZYB ➡ ZUBXYB XAB

AC, AD are diameters.

ABD ABC

ABD +

rt.

ABC 2 rt:

CBD is a st; line.

*(-180′′ — — ABC) 90°-8/2 etc

PROOF: Use a, 0,

represent thes as shovini.

cb.; a = D..

But c+d+CAD=180* b, b2+2 CAD 180°

LESSON 24 : - LOCUS

In a plane, a curve le said to be the Locus or a moving point, if, when the point satisfies a given law, it lies on the curve; and conversely, if when

the point is on the curve, it satisfies the riverí: Law.

At this stage, we consider only two fundamental locus-theorems: (a) perp, bisector locus,

(b) bisector locus.

If the position of a point is given by two distinct laws, it may be possible to construct the points by taking the intersection of the correspond- ing loci.

Example 1 ABC is equilateral. D,E are points on

AC AB resp. such that BE AD. IT BD meets CE at P. find the locus of P.

Solution:

(1) Let. P be a point which satisfies

the given law.

A ABD = A BCE (S.A.S.)

Hence, BEC = ← ADB

*. E,P,D ̧À are concyclic

BPC 180 – LA— 120":

Plies on a segment on

BC containing an angle of 120.

(11) II P' Le an arbitrary point on the above said

segment, Produce BP to cut AC at D'. CP' cuts AB at E Then,

·ZA+ZB'P'D' — 60 +120′′="180" AA, E, P, D' are concyclic

ZBE ̊C ZAD BR Ext. of cyclic quad. AAS

Hence, any pt. on are BPU satisfies the gicen law, From (1) and (11), we conclude that, the locus of F is the segment on EC containing an angle of 120°.

AGHABH 48 in same seg,

comp. of BAE (ABE)

ACF(t, a ACF) ACK

ZAGK 28 in same seg.

AG bisects HGK

Similar for BH and CK(they meet

Then,

Example 2: A and B are fixed points on a given circle;

PQ is a variable chord of the circle of given length; AP, BQ, produced if necessary, meet at R Find the complete locus of R.

const

PROOF:

alt.

PROOF: Draw OMAB ON CD.

Join OG Consider as: OGIE, OGN ABCD OMON AOGA OGN (RHS) MG NG (MB=ND

proved)

Let E G be the mid-pt. of the minor arcs AB, CD; and if M

is the in-centre of ABC... Then

EBEN

Similarly, EA EN

But EB EA DEN

SAEUN IS isoscelės,

Join G, G, the mid-pt. of CD

CEG DEG

EG bisects the vert, 4 of 180s, EMN, Similarly, EGL PQ 1.e: MN / PQ

UNFQ 1s gran

F G H are mid-pt.

Show that:

MNPG 18 a rect

EG A MIN

(1) ADP - A BPH (AAA) (a) dap then DKKP (iii)∞ *p; then KP➡ KA

(iv) K is the mid-pt. of AD.

Q

Solution: (a) In the circle APOBP'Q' (Fig.

AB is a fixed chord, AQB

PQ la of given length, PAQ = 6 = const.

•R = cm fox.const

<AR* B¤«0' + & PLAQ'

⇒(180 - ∞) +6

* 1807 -(∞-const

A, B, B, R ́ ́ are concyclic: (opn.« surp.) And, the focus of R is the whole circle ARER.

(b) In the circle, AQPBQ" F" • (Fig. b)....

¿ARB = ∞<+ B =

<AQ B

PAQ

(180°- ➡) - @

________≈ 180′′*~(~+A ) = const; ABB, R are concyclic

And, the whole circle ARBR' is the locus of R. Consequently, the two circles (from, "a" & "b"). together form the commlete locus of R.

Join PQ, AQ.

PQCg+ 9

PCA+ ABC FPCA+ 2 CAB

CHK

H, K, Q, P are concyclic (Ext of a quad int. one

Const

(Ext 20

Example 3: A and B are fixed points; AP is the tangent from A to a variable circle, centre B. Find the locus of P.

Solution: Join BP..

AP is a tangent from

A to circle B

APB 90′′.

For any circle, centre B,

APB is always of 90

Hence, the locus of P is a circle

diameter AB,

Example 4:As gaven in fig.. ABC is a given triangle.

If PAQ BAC and PQA BCA, find the locus of P

Solution: Joan PB.

ABC APQ.

BAC = PAQ

*BCA

PQA-

ABC = APQ

By sum of ♣,

A, P, B, Q are concyclin

C

(in same

segment

constant.

EXERCISE 24

The sides A, An of the rectangle AuPH: 119 along fixed lines AB, AC, P is a variable point such that PQ-PR=1". Find precisely the lotus of P. How is the locus affected if R may lie on BA prod. produced?

2) Ab are given points, APB 18 a variable rhombus.

Find the locus of P.

CA

3) A variable carcie, centre r, or given radius, passes through a fixed point A. What is the locus of P, (a) if P lies on a plane through A: (b) without this restriction?

A solid, consisting of two together rolls on a plane. centres of the spheres?

5) If the base BC of

position, and if

equal spheres glud What are the loci of uns

ABC 1s given in magnitude, and BAC is given in magnitude. find the locus of the incentre I of ABC-

6) Given an equilateral triangle ABC, construct a point

P on AB such that PA is equal to the perpendicular

PN from P to BC% Give reasons.

門樂百

映仍今滿日連

̇映献天气

座速九點半點三點半點

文中上片·彩七部發

育僑華

1969

英文科 (廿四)

ENGLISH (24)

Answer to Paper XXIII

The big stores of today are not merely ono pas they are a kind of universal exhibition in which we oan, mingle with our fellow-men to marvel at the variety and novelty of the articles exposed for al. The arts of the stores have transformed shopping into a pastime, and the comparatively few people who still profess to dislike shopping must be sither ignorant or extremely unimaginative. Our modern stores can offer all sorts of amenities to the public and are yet able to pay their way. In colour and fascination they rival and even surpass the famous markets of ancient times, and they are likely to be regarded by future generations as one of the wonders of the present age.

Faper XXIV

Put in the correct punctuation Barksi she arrived home to find her mother a large not untidy woman doing some of her hurried cooking the fact that if was a warm day and would be a close evening had not deterred mrs salter from attempting

very hot and heavy meal for she took no notice of the weather but catered by immediate inspiration so that in January the sight offer you cold pork pie and tinned pineapple in august friend steak and a sust pudding this seemed reasonable to all the family exospt rose who was they said a bit hard to please mother she announced at once yes love ive got the sack well you silly girl cried kra maltor lifting her large round rad face from the low oven what ever for rose told the story of the afternoon, as well as she could blaming neither herself nor her employers her mother did not clearly understand what it was all about but by this time she had given up trying to understand all these things every other day one of her children brought home some fantastic tale bui she was neither cross nor worried she was even pleased this was drama of a kind and the salters liked drama

B. Fill in the blanke with the correct preposition 1

a year ago, (1) my way nome (2) snopping, noticed a wild duck flying round frantically, and finally alighting (3) the chimney (4) the house next door, thirty yards away, (5) a whole day more it kept flying round, stopping (6), various chimneys. (7) part (8) three days a woman living (9) a houé (10) the end (11) the street had heard strange noinas (12) which she could not account. When her son called she mentioned this (13) hin, thinking that something was up (14) the attic. A search resulted (15) the finding (16) a wild duck (17) the base (18) a twenty-five- foot-chimney. (19) its attempts to fly up It had worn the wing feathers so badly that it wa unable to fly when it was released. The poor bird WAR a mans (20) soot, hardly recognisable as a bird of any kind.

G. Rewrite the underlined expressions in the

following sentences in simple clear language: (1) I do not think he will set the Thames on fire, (11) I beg to acknowledge the receipt of your

esteemed favour of even date.

He was within an of success.

(111) The only way to treat such a person is to

gend him to coventry

(v) The defenders still held out, hoping against

hops that the king would send help.

(vi) It was evident from their noisy behaviour that

they had dined not wisely but töo well, (vii) When the roll was called, Johnson was

conspicuous by his absence.

(vili) After a long day in the open air, we slept

the sleep of the just.

(ix) Unless you are more careful, you will certainly

be hauled over the coals.

(x) The boy received adequate remuneration for his

services.

D. Give one word for each of the following expressions:

a tendency to believe the worst about people

&

talde, and damaging statement made public

in writing...

(111) the story of a man's life written by himself,

(iv) the absence of all government

(v) the process of recovering after an illness

(vi)

(vii)

a person who is at home in many countries an exclusive trading right

(viii) government by the will of the people

(ix) prove to be true

(x) find to be the sama

斯麗樂永租多 聲寶華金 華英|京英 聲寶華金華英京英

EE

白烈哈"

松激影:公师:「墙四今映祗 塘四命映醒映九點十壎早小 | 映殿一十場早

戰烈形七司类: 片豆牌主蒙拉泯 動能打偵舞曲 夜魂回家世淮秦

演主点白・瑛套|演主辦白·琐覡

一九八新 四三七活歌

邦聯

波

人士 王頓

橋血

光

3

·旺達利斯 忌: 藝忙鮮杯體案無螢

FANTASTIC

SUPERMIN

誠天飛

暖 T

頭

堡戰

雄英 雄英重

【上藝,案地天門探險拆

彩七

彩

刦鑽金 填於公半點五台 | 场除公牛點五分 龍蛟戰百花⁑剛金钱 案大 R 菸

楚黍翼,片面彩七利鴨園

片烈彩。

口街榮福道山青埗水深、 臺灣結天今映只:

張火戰南越

THE

·斯寇尼東

記®

BOSTON STRANGLER 花·西天

GARNEST

垄翻計人美海諜

我巨七璞正下

零片彩球映場期

作茞猛勇彩糖 彩蕺飛:一早映場叫天明 瑞公 長案炒起天明

盛大通帅 傳染多七期下演 門李科為樂大前天晚

DEXIN HEALEYURE.

為一铁四餘火戰南越傳正女飛黠

春雪紅

片蕺色彩的器

伏

賀梅

新白:

·作傑榮尊