Question

A parabola with a vertical axis has its vertex at the origin and passes through point (7,7). The parabola intersects line y = 6 at two points. The length of the segment joining these points is

A. 14
B. 13
C. 12
D. 8.6
E. 6.5

Answer 1

`\boxed{\text{B. 13}}`

Explanation:

1. Find the equation of the parabola

The vertex is at (0, 0), so the axis of symmetry is the y-axis.

The graph passes through (7, 7), so it must also pass through (-7,7).

The vertex form of the equation for a parabola is

y = a(x - h)² + k

where (h, k) is the vertex of the parabola.

If the vertex is at (0, 0),

h = 0 and k = 0

The equation is

y = ax²

2. Find the value of a

Insert the point (7,7).

7 = a(7)²

1 = 7a

a = ⅐

The equation in vertex form is

y = ⅐x²

3. Calculate the length of the segment when y = 6

`\begin{array}{rcl}6 & = & \frac1{7}x^{2\\\\42 & = & x^{2\\x & = & \pm √(42)\\\end{array}`

The distance between the two points is the length (l) of line AB.

A is at (√42, 6); B is at (-√42, 6).

l = x₂ - x₁ = √42 – (-√42) = √42 + √42 = 2√42 ≈ 2 × 6.481 ≈ 13.0

`\text{The length of the segment joining the points of intersection is }\boxed{\mathbf{13.0}}`