1991-12-19 — Page 24

育教

國民華中

日四十月一十年未辛歷夏

|山間爬格記

事 - 令人

賭的故

東京魔儈

12.

報日僑

Let (xq. 4), (X2 Y2) be the coordinates of the two points of

contact.

The equation of the two tangents be

16x4x + 25y+y = 400 .... (1)

1992 中學會考預習專欄

16x2x + 25y2y

= 400

Sub. (10, 0) in (1) and (2)

明德出版社

160x4 - 400

X1

MILL & DALE PRESS

160x2 = 400

(2)

the two points lie on the line x =

Additional Mathematics (11)

Sub. x

in 16x25у* 400

W.S. SO

四期星 日九十月二十(一九九一)年十八國民華中(24)

(24k+ 16k2)

x + x2

*

1+ 4k2

+X

2=-

=

211 + 484)

2 - 4k+ 16k2

2 + 8k2 = 1 - 2k + 8k1

k =

The required equation is

y = -

2x + 2 + 1

x + 2y -6 - 0 (Ans.)

6. The other asymptote is x + 2y =.0.

The hyperbola will be (x+2y) (x - 2y) k.

x2-4y2 k

Sub. (4, 1) 16 - 4 - k

k = 12

The required equation is x2 - 4y2 = 12.

xy = 1

differentiate with respect to x

此誘惑呢?

,戒不禁叫絕。

·難得撥雲見月,那個鶴山一郎,不會走出來吟詩 黑漆的天空 - 雲緩緩移動,露出一個冷月來,人

情軒札記

李天機

今昔觀

Revision Exercise 11 Coordinate Geometry IV: Conics Section A

1. A chord of the parabola y2 = 6x passing through the point P(4,3) is bisected by the point P. Find the equation of this chord.

2. Two tangents drawn from the point P(10, 0): 1

to the ellipse 16x+25y* 400. Find the distance between the two points of contact.

3.

3. Find the equation of the commen tangents to the parabola y2 5x

and circle 9x2 + 9y2 = 16.

4. Find the equation of the two tangents to the ellipse

5x + y2 1 which are perpendicular to the line 2x+5y-2=0.

5. Find the equation of a chord of the ellipse x+4y2-2x-12y+6=0

which the point P(2, 2) is the mid-point.

6. Find the equation of a hyperbola which has a asymptote x-2y=0

and passes through point (4, 1),

7. Find the area of a triangle bounded by the two axis and a

tangent to the heperbola xy^» 1.

Section B

(a) Show that the point Alt,, 2t), B(t, 2t) lie an the

parabola y2 = 4x.

(b) If P(2, 1) is the mid-point of line AB, show that t12-2.

(c) Find the length of the chord AB.

(d) Show that the slope of the line AB is 2, and hence find the

equation of line AB.

Solution:

1. The equation of line through (4, 3) is.

y-3= k(x-4)

Sub, ykx - 4k+ 3 into y2 = 6x

(kx - 4k + 3)=6x7

k2x2- 2(4k 3k+3)x + (3 - 4k)2 = 0

XX are the roots of equation (1)

2(4k2

X1

+

X2

-

~ 3k - 3).

= 4

X4 X2 are the x-coordinates of the two end pos

4.

4k2 - 3k - 3 = 4k*

ki 1

The required equation is y

3

(1).

chard

16(5)2 + 25y* = 400

25y2 = 300

yz 12

ji a ±2√3

The coordinates of the points of contact are

15, 2√3), (5,- 2√3).

The distance between the two points of contact is 2√3+2√3 = 4√3 units.. [Ans.)

Let the equation of common tangent be y = mx +k.

(mx + k) = 5x

m2x2 + (2mk - 5)x + k2 = Q

0

A (2mk - 5)2 - 4m2k2 = 0

-20mk25 = 0

5 K = J3

T

(1)

9x2 + 9{mx + k)2 = 16

(9 + 9m2)x2 + 18mkx +9k2 - 16 = 0.

(18mk) - 4(9 + 9m2)(9k2 - 16) = 0 324m2k2 - 324k* - 324m2k2 + 576m2 + 576 = 0·

-16(m2 + 1)

·k2 =

(2)

From (1), (2)

25 16m2

(m2

256m (16m2 9)(16m2 + 25) = 0

16m2 = 9

+ 256m2 - 225 = 0

16m2 = -25 rejected

= W

m=

k =

The required equations are

+ y = 3 x

4. Let the equation of tangent DE

5x2+

(Ans.)

The gradient of tangent to xy = 1 at (tp:

is

M =

dxlx=t

Equation of tangent:

y

x+ty 2t0

When x = 0, y = (y-intercept)

When y = 0, x = 2t (x-intercept)

Area of triangle =

글(룰)(2)

2 sq. units

8. (a) Sub. (t,, 2t,) in y2 = 4t

L.H.S. = (2tq)2 = 4ty?.

R.H,S. = 4*,2

L.H.S.R.H.5.

Sub. (t, 20) in y2 = 4x

L.H.S. = 4t22 - R.H.S.

8 lie on y=4x

(b) tq?

+

ty

= 4. t + t2 =

+

1

(Ans.)

(Proved)

(Ans.)

圖

(c) (AB) = {t,'

+

(21

(Proved)

2)

4.

2+

2

+

(t,2 +

-

1694(19)

35

+5kx + k − 1 = 0

45x2 + 20kx-+ 4(k2- 1) (20k)* - 4(45)(4)(k2

0

4k2 = 9

K: ±

(Ans.)

(Ans.)

2

(Proved)

Bk - 2) = 0

P(2, 1) lies on AB. Equation of AB:

2

2x y 30

(Ans.)

The required equations are y =

5. The equation of line through (2, 2) is y± kx + 2 - 2k.

x2 + 4{kx - 2 - 2k) - 2x - 12(kx + 2 - 2k) + 60 (1 + 4k2}x2 - (2~ 4k + 16k2)x + (16k2

XX are the roots.

0

-4 = 0

(Ans.)

g. to

on personal taxation or give advice management.

h. to buy and sell securities for clients.

1. to accept valuables for safe-keeping.

investment

J. to issue and sell bankers' drafts, or gifts coupon.

HONG

NG KO

GK

目的擧信

1992 中學會考預習專欄

明德出版社

MILL & DALE PRESS

Commerce (11)

S.K. LAI

香港華商旅遊協會會員聯合廣告

Finance and Banking

Studying Areas

The main studying areas in this unit include:

a. services of banks and Finance companies to the business

firms and personal customers.

b. methods of payment.

c. factors that are promoting Hong Kong to be a financial

centre.

d. recent trends: credit card and automatic teller machines

services.

e. changing retail banking services and its reasons.

11. Services of Banks and Finance Companies to the Business Firms

and Personal Customers

1. The services offered by commercial banks to firms include:

a. Providing professional business advice

Banks offer a wide range of advice:

1.

it.

advice on taxation, borrowing or alternative sources of finance,

advice on investment management, and

iii. advice on economic affairs such as the market

condition of the overseas trading countries.

b. Providing corporate credit card facilities

In order to provide the availability of instant credit to firms, banks issue corporate credit card.

c. Accepting and discounting bills of exchange

d. Underwriting new issues of security

In the capital market, banks underwrite the issue of new corporate shares or debentures.

e. Arranging syndicated loans

In order to arrange syndicated loans for firms, banks raise large fund from various sources,

f. Portfolio management and corporate finance

Bank experts give financial advice on matters of capital- raising fund and investment portfolio management firms.

to

2. The services other than accepting deposits and giving loans

provided by banks to personal customers

The banking services offered by banks to personal customers include:

a., to monitor direct debit and credit transfer systems.

b. to give bankers' references.

c. to issue credit card.

d. to offer ATMs that allow clients to draw cash round the

'clock.

e. to act as executors and trustees.

f. to offer travellers with travellers' cheques and foreign

currency.

3. The criteria of granting a loan by bank to his customers

72

The manager will have to consider the suitability of the

ustomer (applicant).'

cred

character of the customer

manager wi

asthines ID

UBLIC

b. The amount of the Jogh and sthẳ

capability of the customer to repay

thood of

the

It is the manager's job to assess the chance of the customer being able to repay the loan.

c. The availability of the collateral security

In some case, the borrower may even have to find a personal guarantor for the loan in the event of default. d. The risk

Any special risks to be involved in the expansion of the business will have to be considered.

e. The business records

Past business records and present level of business activity should be studied.

The likelihood of its continuing profitability are to be assessed.

f. The perfod of repayment

The perfod for which he needs the loan considered.

g. The purpose of the loan

The loan to be used should be assessed.

4. Differences between a bank loan and an overdraft

is to be

a. A loan is for a stated sum of money for a fixed period of

time.

An overdraft occurs when the holder of a current account is allowed to draw cheques for more than the holder has in his account, up to an agreed maximum,

b. The interest charged is fixed for the whole length of the Toan. For the overdraft, the interest charged is based on the amount overdrawn on a daily basis.

c: To a loan, it is used for a special purpose. To an

overdraft, it is for temporary relief.

When the customer needs cash, he may overdraw the account.

5. Deposit-taking Institutions in Hong Kong

With

effect from 1 February 1990, the deposit-taking institutions have been divided into three new distinct classes:

a. Licensed banks,

b. Restricted licence banks, and

c. Deposit-taking companies.

The main features among these financial institutions are compared and contrasted as follows:

Similarities

a. They have to comply with the provisions of the Banking

Ordinance.

b. They maging in foreign exchange dealings, lending,

and deposit-taking.

They are diversifying into other financial services: invested advice and fund management.

Differences

Paid-up share capital

IBR

Licensed banks, they must have at least $150 million.. For restricted licence banks, they are required to have a minimum of 100 million.

For deposit-taking companies, they must have a minimum of $25 million.

b. Deposits

For licensed banks, they may take deposits of any size and any maturity from the public.

to

For restricted licence banks, they may accept deposits of any maturity with amounts of not less than $500,000. For deposit-taking companies, they are restricted accepting deposits of not less than $100,000 with an original term to maturity of at least three months. c. Status-granting

For licensed banks, they are granted at the discretion of the Governor in Council under Banking Ordinance.

For restricted licence banks, they are granted at the discretion of the Financial Secretary.

For deposit-taking comparies, the authority rests with the Commissioner of Banking.

d. Current account

Only licensed banks may operate current account.

III. Methods of Payment

We can pay for goods and services in any one of the following methods:

a. by cash,

b. through the commercial bank, or

c. through the Post Office.

1. Cash

Payment can be made by cash in the form of coins and notes in the most business transactions. It is one of most familiar and convenient forms of payment.

However, it is inconvenient to deal with large sums of monetary transactions as it is too bulky to carry around.

2. Cheque

A cheque is a bill of exchange drawn on a banker payable on demand.

2.1 Advantages of making payment by cheque

The advantages are described below:

a. It is much safer than using cash, thus reducing chances of theft. Moreover, the account holder of cheque can cancel the cheques if a cheque or cheque book is lost.

b. A cheque can be made out for any amount, thus

promoting the flexibility of a cheque.

c. It is easy to carry around a cheque or a cheque book whereas large sums of cash can be too bulky and inconvenient to despatch.

d. Cheques can be used as proof and record of payment since a drawer can use his bank statement to find out whether his bank has honoured his cheques.