CPhill

Thanks, EP.....Here's another way to see this :

If

1  = ( x + 12) / (x - 4)....then the numerator and denominator would have to be the same thing because   a / a  = 1    [ and a ≠ 0 ]

LOL!!!!!

Sometimes......

Logic >  Algebra  ........

(1)

x^2 + y^2  = 5    (1)

y  = 2x + k   (2)

The slope of the line is 2......usiing implicit differentiation...the slope of any tangent line to the circle is given by    ....    -x / y

So  ....equating  slopes

-x / y =   = 2     →   x  = -2y         (3)    ...... sub this into (1)

(-2y)^2 + y^2  = 5

4y^2 + y^2  = 5

5y^2  = 5     →   y  = ± 1

When y = ± 1 ....using (3),  x = - 2  or x = 2

So....we have the points      (-2, 1)  and (2, - 1)

And using (2) and solving for k........  k  =  5      or k = - 5

Here's a graph :   https://www.desmos.com/calculator/qkhdev0shy

(2)

And it's obvious that the line will intersect the circle at two points whenever 

-5 < k < 5

(3)

And  the line will not intersect the circle whenever 

k < - 5   or   k > 5

Let your broher's age  = 10a + b    where a and b are the digits of his age 

And your age is      10a + b - 25  =  10(a - 2) + (b- 5)   where (a -2) and (b- 5)  are the digits of your age

And....the digits of his age multiplied togeher = your age

ab = 10a + b - 25     (1)

And.....the sum of he digits of your age  = the first digit of his age

(a - 2) + (b - 5)  = a  →  a + b - 7  = a  → b - 7 = 0 →   b = 7

So ......using (1) we have

a7 = 10a + 7 - 25    →

-3a  = - 18    →

a  = 6

Your brother  is 67 and you are 67 - 25  =  42

For a function f(x), it's reciprocal is defined as 1 / f(x)

So  if

f(x)  = 1/ (√(x^2  –  1)

The reciprocal function is :

1 / f(x)  =  1 / [ 1/ (√(x^2  –  1) ] =   √(x^2  –  1)

It's settled......30 machines......  a bottle for all the answerers......

If 5 machines can make 1,000 bottles in 4 days......they can make 1/4 of this =  250 bottles in one day

So....to make 1500 bottles in one day, we need :

1500/250  * 5  =

6 * 5  =

30  machines

Yes....always.......

Hey...don't get smart with me, young lady......!!!!

P.S.  ......I'd never figure out how you did this...but...it should be worth more than 5 points....!!!