Parabola a can be represented using the equation (x + 3)2 = y, while line b can be represented using the equation y = mx + 9. isabel determines one solution to the system of two equations must always pass through the vertex of parabola a. which best describes the reasonableness of her solution? a: isabel is correct because the y-intercept of line b is (0, 9) and the value of y when x = 0 in parabola a is 9. b: isabel is correct because the y-intercept of line b is (9, 0) and the value of x when y = 0 in parabola a is 9. c: isabel is incorrect because the point of intersection between line b and parabola a that can be determined is the y-intercept of the functions, not necessarily the vertex. d: isabel is incorrect because the point of intersection between line b and parabola a that can be determined is the x-intercept of the functions, not necessarily the vertex.

We have that
y=(x + 3)²
y=mx+9

we know that

the y intercept of the Line is for x=0
then
y=m*0+9---------> y=9

and the y intercept for the parabola is
for x=0
y=(x + 3)²-------> y=(0 + 3)²--------> y=9
so 
both y intercepts of the two functions coincides in the same point
the point (0,9) determines one solution to the system of two equations

the y-intercept of the functions, not necessarily is the vertex.
In this problem the vertex of the parabola is the point (-3,0)
see the attached figure

the answer is the option
C: Isabel is incorrect because the point of intersection between Line B and Parabola A that can be determined is the y-intercept of the functions, not necessarily the vertex

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12