Question

An air show is scheduled for an airport located on a coordinate system measured in miles. The air traffic controllers have closed the airspace, modeled by a quadratic equation, to non-air show traffic. The boundary of the closed airspace starts at the vertex at (10, 6) and passes through the point (12, 7). A commuter jet has filed a flight plan that takes it along a linear path from (–18, 14) to (16, –13). Which system of equations can be used to determine whether the commuter jet’s flight path intersects the closed airspace?
StartLayout Enlarged Left-Brace 1st Row y = one-fourth (x minus 5) squared + 10 2nd Row y = negative one-half x + 5 EndLayout
StartLayout Enlarged Left-Brace 1st Row y = one-fourth (x minus 5) squared + 10 2nd Row y = negative 2 x minus 22 EndLayout
StartLayout Enlarged Left-Brace 1st Row y = one-fourth (x minus 10) squared + 6 2nd Row y = Negative StartFraction 27 Over 34 EndFraction x minus Five-seventeenths EndLayout
StartLayout Enlarged Left-Brace 1st Row y = one-fourth (x minus 10) squared + 6 2nd Row y = negative 2 x minus 22 EndLayout

Answer 1

First find the parabola. Vertex (10,6) and passing through (12,7) gives
7 = a(12−10)^2 + 6 → 7 = 4a + 6 → a = 1/4,
so y = (1/4)(x − 10)^2 + 6.

Find the line through (−18,14) and (16,−13): slope m = (−13 − 14)/(16 − (−18)) = −27/34. Using (−18,14):
b = 14 − (−27/34)(−18) = 14 − 243/17 = −5/17,
so y = −(27/34)x − 5/17.

Thus the correct system is
y = (1/4)(x − 10)^2 + 6
y = −(27/34)x − 5/17

(which is the third option).