CPhill

First....let A be located at (0,0)....construct a circle with radius 1 and centered at A

And this circle wil pass through P  and have the equation

x^2 + y^2  =  1  →  y^2  =  1 - x^2   (1)

And let B be located at (c,0).........construct a circle with radius 7 and centered at B

And this circle will pass through P and have the equation

(x - c)^2 + y^2 = 49          sub (1) in  for y^2  

(x -c)^2  + [1 - x^2 ]  = 49  

(x - c)^2  = 48 + x^2   (2)

Next......let C  be located at (c,d).........construct a circle with radius 8 and centered at C

And this circle will pass through P and have the equation

(x - c)^2 + (y - d)^2  = 64        sub (2) in for  (x - c)^2  

[48 + x^2]  +  (y - d)^2  = 64  

(y - d)^2 = 16 - x^2      (3)

Finally......

Let D  be located at (0,d).........construct a circle with radius DP and centered at D

And this circle will pass through P and have the equation

x^2 + (y - d)^2  = DP^2       sub (3) in for (y - d)^2 

x^2 + [ 16  - x^2]  = DP^2    .......  simplify..........

16 = DP^2        take the positive root

4  = DP =  PD

The major axis will lie on the y axis and the center is (0,0)

"a"  = 5     and  "c" = 3

And we need to find "b"  in this equation

y^2 / 25   +  x^2 / b^2   =   1

b  = sqrt (a^2 - c^2)  = sqrt ( 5^2 - 3^2)  = sqrt (16)  = 4

So......b^2   =  16

And the equation is

y^2 / 25 + x^2 / 16  = 1

Here's the graoh : https://www.desmos.com/calculator/kzjn2slwcp

Thanks, X².....your method is easiest.....

Here's an algebraic solution.....

The x coordinate of the vertex  is    -b/ [ 2a]....so we have

-b/ [2a]  = -6

b = 12a

And we know that

a(-6)^2  + 12a(-6)  + c   =  -14

36a  -72a + c   = -14

c = 36a - 14

And since 17  is a root.......we have that

a(17)^2  + 12a (17)  +  [ 36a - 14 ] =  0

289 a + 204a  + 36a  = 14

529a  =  14

a  = 14/529

b = 12a  = 168/529

c = 36 a - 14  = -6902/529

So.......our function is

f (x)  =  (14/529)x^2  +  (168/529)x - 6902/529

Set this to 0   and multiply through by 529 and we have that

14x^2   + 168x  - 6902  =  0      divide through by 14

x^2  + 12x - 493  = 0   and this factors as

(x - 17) (x + 29)  = 0

Setting the two linear factors to 0 and solving for x gives the two roots  x = 17  and x = -29

Just as X²  found......!!!!!

Nobody knows.......most of our answers on here are only rough guesses......

Look at the right side

   1 / (x - 1)  + 1

Note that we can write 1  as    (x - 1) / (x -1)

And  we have

1 / (x -1)  + (x -1) / (x - 1)     so we have a common denominator

So we have

[   1   + ( x - 1) ]  / ( x - 1)  =

[ 1  +  x  - 1 ] / ( x - 1)  =

x / (x - 1)  = the left side

The distance that the bus travels each minute is  42/60 miles.....and it has a 15 minute head start

So....in 15 minutes....the distance that your mother will have to close to catch the bus is     

(15) * 42/60  miles  

The distance that your mother travels each minute is 72/60 miles

So......the time it will take to catch the bus is given by

( 72/60 - 42/60) T  = 15 * 42 / 60

(30/60)T = 15* 42 .60      multiply through by 60

30T   =  15*42   divide both sides by 30

T = (1/2) 42

T  = 21  minutes

So.....15 minutes after 7:18 AM is 7:33AM....and 21 minutes after this is 7:54 AM....and this is when she will overtake the bus

Your explanation is just fine, Gh0sty.....!!!!

As Melody has said, "theta"  is a 4th quadrant angle

tangent  = y / x  =    -1    .... so......x = 1     and   y  =  - 1

And  r =   sqrt (x^2 + y^2)  =  sqrt [ 1^2 + (-1)^2 ]  = sqrt (2)

Sine  =  y / r  =  -1 / sqrt (2)

Cosine  =  x / r  =   1 /  sqrt(2)

OOPS...sorry.....!!!!!

I didn't look at it .....I thought it was the same question I had answered elsewhere....

My bad !!!!

\($\frac{a+b}{a-b} + \frac{a-b}{a+b}$\)

[ (a + b)(a + b) + (a - b)(a - b) ]  / [ a^2 - b^2 ] =

 ( [ a^2 + 2ab + b^2  + a^2 - 2ab +b^2 ] / [ a^2 - b^2]  =

[ 2a^2 + 2b^2 ] / [ a^2 - b^2 ]

2 [ a^2 + b^2] / [ a^2 - b^2 ]

The smallest positive value, 2, is achieved when  b  = 0  and a is any positive integer

In every other case,  [ a^2 + b^2] / [ a^2 - b^2 ] will be > 1   and the expression will be > 2