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In the equation of the parabola below, the coefficient of $x^{2}$ is $1$ .

The area of the given rectangle is $1458$ .

Find the length of the line segment $O P$ , giving your answer as a fully simplified surd in the form

a \sqrt{b}, a, b \in N

Give the value of $a b$ .

Hint

Since the parabola has a single, positive root, it must be of the form

y = (x - k)^{2}

for some number $k > 0$ .

Solution

Since the parabola has a single, positive root, it must be of the form

y = (x - k)^{2}

for some number $k > 0$ .

By symmetry, $P$ must occur when $x = 2 k$ , and so $y = (2 k - k)^{2} = k^{2}$ .

Since the area of the rectangle is given, we have

\begin{aligned} 2 k \times k^{2} & = 1458 \\ k^{3} & = 729 \\ k & = 9 \end{aligned}

Thus, we find that $P (18, 81)$ and, by Pythagoras,

\begin{aligned} O P^{2} & = 18^{2} + 81^{2} \\ = 9^{2} (2^{2} + 9^{2}) \\ = 9^{2} \times 85 \\ O P & = 9 \sqrt{85} \end{aligned}