CPhill

Welcome aboard   !!!!!

Thanks, Dimitristhym.......!!!!

In the equation of the circle.....sub   2x + 11   for y......and we have

( x + 5)^2  +  (  2x + 11 - 6)^2   =  25      simplify

x^2 + 10x  +  25   + ( 2x + 5)^2   =  25

x^2 + 10x + 25 +  4x^2 + 20x + 25  = 25

5x^2 + 30x + 50  = 25        subtract 25 from both sides

5x^2 + 30 x + 25   = 0         factor out 5

5 ( x^2 + 6x + 5)   =  0       divide both sides by 5

x^2 + 6x + 5   = 0

Factor  x^2 + 6x + 5  =  0     as

(x + 5) ( x + 1)  = 0

Set each factor to 0  and solve for x and we have

x = - 5      and  x  =  - 1

These are the x coordinates of the intersections...and the y coordinates are

y = 2(-5) + 11  = 1        and    y = 2(-1) + 11  =  9

So...the intersection points are

(-5, 1)   and ( -1, 9)

Correct.....the cosine  and sine   are negative in Q3   !!!!!

We know that   x  = 8  and y = -7

r  =  sqrt  [  (8^2   + (-7)^2 ]   =   sqrt [ 64 + 49]  =  sqrt [ 113]

So.....cos (theta)   =  x / r   =   8 / sqrt (113)

Multiply both numerator and denominator by  sqrt [ 113]   and we have

8 sqrt (113)                             8sqrt (113)

_________________ =         __________

sqrt(113) * sqrt (113)                 113

Thanks, tertre   !!!

2.

P dilated by a factor of  (1/3) about the origin  =   (12, 6) (1/3)  =  (4, 2)

Q dilated by a factor of 1/3 about the origin  =  (-9, 15)(1/3)  =   ( -3, 5)

No prob, RP  !!!!

Let's work with this, first

  2               1

___    +     ___

x - 1            x

Cross-multiply  and keep the same sign as we have between the fractions

2x +   1 * ( x  - 1)   =   2x  + x - 1   =    3x - 1

This wiill be the numerator of the fraction at the top

Take the product of the denominators = x ( x - 1)

This will be the denominator of the fraction at the top

So.....we have

3x - 1                8

______   ÷       ____                invert the second fraction  and multiply

x(x - 1)               x

3x - 1                x                   x ( 3x - 1)

_______    *  _____  =      ____________

x ( x - 1)           8                 8x ( x - 1)

Cancel the x's on top/bottom

(3x - 1)

______

8 (x - 1)

Note that we have a common denominator at the top....so this is just

[ 1 - x^2  ] / 4

Notice that  x  can be wriiten as    x  / 1

So  we have

[ 1 - x^2 ] / 4   ÷  [ x / 1 ]

Invert the second fraction and multiply

( 1 - x^2)               1                   (1 - x^2)

_______    *      ____   =          _________

      4                    x                        4x