Question

a straight line with negative slope passes through the point (9,4) and cuts the positive co ordinate at P and Q respectively
i) min value of OP+OQ (where O is origin)
ii)Area of OPQ when OP+OQ is min

Answer 1

options for
i)
18
25
36
49

Answer 2

options for
ii)
75
225
125
275

Answer 3

looks like calc to me yes?

Answer 4

we were asked this in co ordinate geo test paper!!

Answer 5

the points are say (0,y) and (x,0) and you know they have to pass through (9,4) so the slopes must be the same. that way you can write one in terms of the other.

Answer 6

really? find the max w/o calc? ok lets see

Answer 7

slope from (9,4) to (x,0) is
\[\frac{-4}{x-9}\] and slope from
(0,y) to (9,4) is
\[\frac{y-4}{-9}\] and these have to be equal so i guess
\[\frac{-4}{x-9}=\frac{y-4}{-9}\] maybe we can solve for one in terms of the other

Answer 8

i tried to assume a slope m
y-4=m(x-9)
x intercept=9-4/m
and
y intercept=4-9m

Answer 9

OP+OQ=9-4/m+4-9m
=9m-4+4m-9m^2
----------------
m
m is negative

Answer 10

oh this is probably better than what i was trying
\[x=9-\frac{4}{m}\]
\[y =4-9m\]

Answer 11

-9m^2+13m-4
------------- needs to be minimum
m
do u think differentiating it would be the best way possible?

Answer 12

wait i got lost somewhere there. when you add you get
\[9-\frac{4}{m}+4-9m\] yes?

Answer 13

oh ok i see you put it all over m
\[\frac{-9m^2+13m+4}{m}\]

Answer 14

ok and we want the minimum value out of this. that will give the slope. looks like you are doing better than me.

Answer 15

except i think it should be +4 yes, not -4 or did i mess up?

Answer 16

oh no it is -4

Answer 17

Can't you just look at this geometrically?

You have a line pivoting about (9,4) through pi/2

Answer 18

u mean the angle wid x axis as pi/2?

Answer 19

ok derivative is
\[\frac{4-9x^2}{x^2}\] set equal zero get
\[9x^2=4\]
\[x=-\frac{2}{3}\]

Answer 20

since x is negative.

Answer 21

From min to max, the slope (Tan) varies from y=9+, x is 4+

Answer 22

put in
\[-4=-\frac{2}{3}(x-9)\] gives
\[x=15\]

Answer 23

so x intercept is (15,0) and then y = 10

Answer 24

wow you had it. better than my method.

Answer 25

can u jst show the working of how u got the dervatve coz i m getting a cubic differentiating it

Answer 26

sure but why don't we do it the easy way first?

Answer 27

we actually started with
\[-9m+16-\frac{4}{m}\] before we put over one denominator. so just take the derivative of this thing

Answer 28

get
\[-9+\frac{4}{m^2}\] right away

Answer 29

oh
and i wasted 15 minutes in the paper breaking my head over the cubic!!!!!

Answer 30

or
\[\frac{4-9m^2}{m^2}\] which is what you will get if you use the quotient rule. want to do it that way too?

Answer 31

yeah but it will work out the same with more algebra for sure. can we skip that or would you like to do it as well? up to you

Answer 32

i would love to see the algebric way of solving it

Answer 33

well first of all you do not get a cubic. we have
\[f(m)=-9m^2+13m-4\]
\[f'(m)=-18m+13\]
\[g(m)=m\]
\[g'(m)=1\] and use
\[\frac{f}{g}'=\frac{gf'-f'g}{g^2}\]

Answer 34

gives
\[\frac{(-18m+13)-1(-9m^2+13m-4)}{m^2}\]

Answer 35

ok that was a mistake

Answer 36

\[\frac{m(-18m+13)-1(-9m^2+13m-4)}{m^2}\] is better

Answer 37

numerator is
\[-18m^2+13m+9m^2-13m+4=-9m^2+4\]

Answer 38

yep u got it

Answer 39

and that gets it

Answer 40

thanks a lot

Answer 41

someone help me lol please :) you guys are really smart !

Answer 42

but you did all the work. the rest was mechanics. solution is x = 15, y = 10 and done

Answer 43

@tabby choice c

Answer 44

to tabbeyx33 it is the 3rd option

Answer 45

thanksthanks <3333 lov yaa :)

Answer 46

so let me scroll up and see what we got, because if it is not there i am going to be bummed.

Answer 47

yup 25 for the sum and 75 for the product. good work

Answer 48

so last one guys :) its another type of question as the previous i just asked

Answer 49

b

Answer 50

thank you soo much <3 i passed :) wow your like super awesome :)

Answer 51

b it is

Answer 52

yw

Answer 53

Still puzzling over the geometry of this.

In particular, why is it that the slope for the minimum squared = 4/9?

Not a happy coincidence, surely.

Answer 54

If we let y be the difference beween the y-intercept and 9 and x the difference between the x-intercept and 4, we have tangent (slope) = y/4 = 9/x

We want Min(x + y +13) = Min(x + 36/x +13)

Diff = 1 - 36/x^2 -> x = 6 (-6 not) which is evidently the same if you use y instead.

So you have the same 10 by 15 axes reversed.

Answer 55

no i was fairly sure it was not a coincidence that the slope turned out to be 2/3