Question

The axis of a parabola is parallel to the y-axis and its vertex is at \((5, 0)\). If it passes through the point \((2, 3)\), then its equation is:

• A. \(y^2 = 3(x - 5)\)
• B. \(3y = (x - 5)^2\)
• C. \(3y^2 = x - 5\)
• D. \(y = 3(x - 5)^2\)
• E. \(y = 9(x - 5)^2\)

Hint:

Always substitute a known point into the vertex form of a parabola to solve for the coefficient \(a\), which dictates the width and direction of the parabola.

Answer 1

The Correct Option is B

Given the vertex of the parabola \((5, 0)\) and the axis is parallel to the y-axis, the standard form of the equation of the parabola is: \[ y = a(x - h)^2 \] where \((h, k)\) is the vertex. Here, \(h = 5\) and \(k = 0\), so: \[ y = a(x - 5)^2 \] We know the parabola passes through the point \((2, 3)\). Substituting \((x, y) = (2, 3)\) into the equation gives: \[ 3 = a(2 - 5)^2 \] \[ 3 = 9a \] \[ a = \frac{1}{3} \] Therefore, the equation of the parabola is: \[ y = \frac{1}{3}(x - 5)^2 \] Multiplying both sides by 3 to match the answer format: \[ 3y = (x - 5)^2 \]