The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B. The mid-point of AB is the point C. Show that the point C lies on the line x + y = 11.

Please see the attached picture for the full solution.

9514 1404 393

Explanation:

Substituting the cut line into the curve equation, we have ...

  x(x+10) = 11x +5

In standard form this is ...

  x^2 -x -5 = 0

The sum of the two roots of this equation is the opposite of the x-coefficient*, so is -(-1) = 1. Then the midpoint between the points of intersection has an x-coordinate of ...

   (root sum)/2 = 1/2

__

The midpoint y-coordinate will be the y-value on the cut line, so is ...

  y = x + 10 = (1/2) +10 = 10.5

Then the midpoint of AB is C(0.5, 10.5). This is easily shown to lie on the line ...

  x + y = 11

  0.5 + 10.5 = 11

* The factored form of a monic polynomial can be expanded to show this:

  (x -a)(x -b) = x^2 -(a+b) +ab

That is, the sum of the roots a and b is the opposite of the x-coefficient.

just take the area of the square and then the area of the circle then subtract the circle area from the squares area

step-by-step explanation:

the polynomial is a perfect square trinomial of the form a^2-2ab+b^2:

option a. true

step-by-step explanation:

9x^2-12x+4

comparing with the form a^2-2ab+b^2

1.) a^2=9x^2

solving for a: square root both sides of the equation:

sqrt(a^2)=sqrt(9x^2)

a=sqrt(9) sqrt(x^2)

a=3x

2.) b^2=4

solving for b: square root both sides of the equation:

sqrt(b^2)=sqrt(4)

b=2

to be a perfet square trinomial 2ab must be equal to 12x. let's check:

2ab = 2 (3x) (2)

2ab = 12x

correct