Stumped... (with occaisional updates) (1 Viewer)

[FDownes]

Now I know this question has a simple solution, but I can't for the life of me remember what it is... Could anyone help me out and demonstrate how to solve it?

The circle x^2 + y^2 - 4x - 6y + k = 0 touches the x-axis. Find k and the coordinates of the point of contact.

Also, if I happen to run in to any trouble in the near-future, I'll use this thread (so I'm not cluttering up the forums with a hundred useless threads ).

 

[ssglain]

The question tells us that the x-axis, i.e. the line y = 0, is a tangent to the circle x² + y² - 4x - 6y + k = 0. What you need to do is solve the two simulaneously whilst noting that tangential behavious implies discriminant = 0. The solution is behind the spoiler.

Spoiler

Substitute y = 0 into x² + y² - 4x - 6y + k = 0 --> x² - 4x + k = 0
Discriminant = (-4)² - 4k = 0
i.e. 16 - 4k = 0
.: k = 4

Solve for x from x² - 4x + 4 = 0
(x - 2)² = 0
.: x = 2
.: Point of contact (2, 0)

Also I would recommend starting a new thread if you need help in the future because older threads tend to get overlooked.

 

[FDownes]

Ah, now I see my problem; I was trying to solve this as though it INTERCEPTED the x-axis. This makes much more sense.

Thanks.

 

[FDownes]

I have a new question, this time about sums to infinity and recurring decimals. I get the basis of the question, but how should I respond when presented with a question like...

Write 0.233333... (where the 3 is repeated infinitely) as a fraction.

I understand that 0.11111... would equal 1/100 + 1/10000 + 1/1000000 and so on, but I'm not sure how to rewrite this decimal so that the two is included in the series. Can anyone help?

 

[pLuvia]

Consider x=0.233333333
then 100x=23.3333333
Solve for x
100x-x=23.33333..-0.2333333
99x=23.1
x=23.1/99
=7/30

 

[FDownes]

I actually figured this out before you got in with your solution, but thanks very muchly anyways. I'm well acquainted with the method you suggested, but the question actually requires the solution to be figured out using the infinity sum of series, so I couldn't use it.

I didn't really understand the method; I was attempting to solve this as though you had to turn the decimal in to a single, repeating series, two inclusive. Instead, you break it up in to two problems...

0.233333... = 1/5 + (3/100 + 3/1000 + 3/10000 + ...)

So you simply the the sum to infinity of 0.033333... and then add it to 0.2, which of course, gives you 7/30.

 

[pLuvia]

Doing it the series way
0.233333..=1/5+(3/100 etc..)
the bracket bit is just a GP so
3/100*(1-(1/10)^infinity)/(1-(1/10))=1/30
So 0.2333333..=1/5+1/30=7/30

 

[kony]

i think the discriminant method is inherently flawed, in that when the thing you're doing it on is not a function, x-discriminant method fails (without careful reasoning), and similarly if it is not a one to one function, the y-discriminant method sometimes fails.

here the reasoning of why x-discriminant method works is simple enough, but here's an easier method:

The circle x^2 + y^2 - 4x - 6y + k = 0 touches the x-axis. Find k and the coordinates of the point of contact.

rewrite the equation as (x-2)² + (y-3)² = 13-k

therefore, the centre is at (2,3) and radius is at root(13-k).

since it is 3 units from the centre down to the point of contact, the radius must be 3. so 13-k = 9, k=4.

 

[FDownes]

Okaaay... Two more maths questions for you. I've made a pretty good start on both of them, but I just can't seem to come to a conclusion.

Question 1:

Find the least number of terms for which the series 20 + 4 + 4/5 + ... is greater than 24.99.

I'm pretty sure I have to use the formula Sn = n/2 x [2a + (n - 1)d] for this one (by subbing in Sn > 24.99 and eventually finding n) but it kind of falls apart when I attempt it...

Question 2:

The sum of the first five terms of a geometric series is 77, and the sum of the next 5 terms is -2464. Find the fourth term of the series.

Again, this is another case of simply using a formula and finding pronumerals (only this time involving a simultaneous equation), but I just can't seem to solve it.

So, can anyone help me out here?

 

[ssglain]

Q1. You're on the right track. The only problem is that you used the formula for sum of AP. The series you have there is clearly a GP with a = 20, r = 1/5.

Q2. Are you sure you used the correct formula?

Edit: In case you forgot, this is the formula: S = [a(r^n - 1)]/(r - 1)

 

[FDownes]

Whoops... There's my problem.