What fraction of the area of the ellipse lies…

What fraction of the area of the ellipse lies inside Q?

Question 113

HardOriginal question

The ellipse $\frac{x^{2}}{36} + \frac{y^{2}}{16} = 1$ meets the axes at $(\pm 6, 0)$ and $(0, \pm 4)$ . Joining these four points in turn gives a quadrilateral $Q$ inside the ellipse.

What fraction of the area of the ellipse lies inside $Q$ ?

\frac{1}{\pi}

Correct answerYour answer

1 - \frac{2}{\pi}

Correct answerYour answer

\frac{1}{2}

Correct answerYour answer

\frac{2}{\pi}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\frac{48}{24\pi} = \frac{2}{\pi}$ .
For any ellipse, $Q$ has area $\frac{1}{2}(2a)(2b) = 2ab$ and the ellipse $\pi ab$ , so the fraction is always $\frac{2}{\pi}$ .

Worth noting

$1 - \frac{2}{\pi}$ is the fraction outside $Q$ .

Concept: Comparing two areas
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsEllipses

Practise this chapter →

Ten questions from Area of Bounded Regions, at your own pace.

Practise this chapter →

Ten questions from Area of Bounded Regions, at your own pace.

What fraction of the area of the ellipse lies inside Q?

Question 113

Explanation