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Chapter 12: Problem 1075

The area enclosed by the parabola \(x^{2}=4\) by and its latusrectum is \((4 / 3)\) then \(b>0\) is equal to....... (a) 2 (b) \(\sqrt{2}\) (c) 1 (d) 4

Short Answer

Expert verified
The value of \(b > 0\) is equal to \(\boxed{\sqrt{2} \ (b)}\).

Step by step solution

01

Determine the vertex and focus of the parabola

The given equation of the parabola is \(x^2 = 4\). We can rewrite this equation as \(x^2 = 4(1)\), which means the vertex of the parabola is at the origin \((0,0)\), and the focus is on the right side at the point \((1,0)\).
02

Find the equation of the latus rectum

The latus rectum is a horizontal line segment that passes through the focus and is perpendicular to the axis of symmetry of the parabola. Since the vertex is at the origin and the focus is on the right side at the point \((1,0)\), the axis of symmetry is the x-axis, and the latus rectum is a horizontal line with a y-coordinate of 0. Therefore, the equation of the latus rectum is \(y = 0\).
03

Find the enclosed area

We will now set up an integral to find the area enclosed by the parabola and the latus rectum. Since the parabola is defined by the equation \(x^2 = 4\) that can be written as \(y = \sqrt{4x}\), we can integrate this equation from \(x=0\) to \(x=b\) with respect to \(x\), and then multiply the result by 2 to find the area of the entire enclosed region. So we have: \[Area = 2 \int_{0}^{b}\sqrt{4x}dx\]
04

Solve the integral

Let's now compute the integral and solve for \(b\). \begin{align*} Area &= 2 \int_{0}^{b}\sqrt{4x}dx\\ &= 2\left[ \frac{2}{3}(4x)^{\frac{3}{2}} \right]_{0}^{b}\\ &= \frac{8}{3}(b^3 - 0^3) \end{align*} We are given that the area enclosed is \(\frac{4}{3}\), so we can set our integral equal to this value and solve for \(b\). \begin{align*} \frac{4}{3} &= \frac{8}{3}(b^3)\\ \Rightarrow b^3 &= \frac{1}{2}\\ \Rightarrow b &= \sqrt[3]{\frac{1}{2}} \end{align*} Simplifying the cube root, we get \(b = \sqrt[3]{\frac{1}{2}} = \sqrt{2}\). So, \(b > 0\) is equal to \(\boxed{\sqrt{2} \ (b)}\).

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