Question

19 mark for review in the xy-plane, the graph of the equation $y = -x^2 + 9x - 100$ intersects the line $y = c$ at exactly one point. what is the value of $c$? a $-\frac{481}{4}$ b $-100$ c $-\frac{319}{4}$ d $-\frac{9}{2}$

Explanation:

Step1: Set equations equal

Set \( -x^2 + 9x - 100 = c \), rearrange to \( x^2 - 9x + (100 + c) = 0 \).

Step2: Discriminant for one solution

For one intersection, discriminant \( D = 0 \). \( D = (-9)^2 - 4(1)(100 + c) = 0 \).

Step3: Solve for c

Simplify: \( 81 - 400 - 4c = 0 \) → \( -319 - 4c = 0 \) → \( c = -\frac{319}{4} \).

Answer:

C. \( -\frac{319}{4} \)