Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 15: Problem 1434

The radius of the circle passing through the foci of the ellipse \(\left(\mathrm{x}^{2} / 16\right)+\left(\mathrm{y}^{2} / 9\right)=1\) and having its centre \((0,3)\) is \(\ldots \ldots\) (a) 4 (b) 3 (c) \(\sqrt{12}\) (d) \((7 / 2)\)

Short Answer

Expert verified
The radius of the circle passing through the foci of the ellipse \(\left(\frac{x^2}{16}\right)+\left(\frac{y^2}{9}\right)=1\) and having its center at \((0, 3)\) is \(4\).

Step by step solution

01

Find the foci of the ellipse

The equation of the ellipse is given by \(\left(\mathrm{x}^{2} / 16\right)+\left(\mathrm{y}^{2} / 9\right)=1\). This is in the standard form for an ellipse with horizontal major axis: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 16\) and \(b^2 = 9\). The distance between the center of the ellipse and each focus is given by \(c = \sqrt{a^2 - b^2}\). Calculate the value of \(c\): \(c = \sqrt{16 - 9}\) \(c = \sqrt{7}\) Now, we can find the coordinates of the foci. Since the ellipse has a horizontal major axis and is centered at the origin, the foci are located at \((-c, 0)\) and \((c, 0)\), or \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\).
02

Find the radius of the circle

The circle has its center at (0, 3), and we just found the coordinates of the foci to be \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\). Since the circle passes through both foci, the distance from its center to each focus is the radius. We will use the distance formula to find this radius: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) Calculate the distance between the center of the circle and one of the foci, for example, \((\sqrt{7}, 0)\): \(r = \sqrt{(0 - \sqrt{7})^2 + (3 - 0)^2}\) \(r = \sqrt{7 + 9}\) \(r = \sqrt{16}\) \(r = 4\) So, the radius of the circle is 4, which corresponds to choice (a). #Answer#: (a) 4

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The line \(\mathrm{y}=\mathrm{c}\) is a tangent to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) if \(\mathrm{c}\) is equal to (a) a (b) 0 (c) \(2 \mathrm{a}\) (d) None of these

The coordinates of a point on the hyperbola \(\left(x^{2} / 24\right)-\left(y^{2} / 18\right)=1\) which is nearest to the line \(3 x+2 y+1=0\) are (a) \((6,-3)\) (b) \((6,3)\) (c) \((-6,3)\) (d) \((-6,-3)\)

Two tangents to the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}=4\) at the points \(\mathrm{A}\) and \(\mathrm{B}\) meet at \(\mathrm{P}(-4,0)\). The area of the quadrilateral \(\mathrm{PAOB}\), where \(\mathrm{O}\) is the origin is (a) \(4 \sqrt{3}\) (b) 4 (c) \(6 \sqrt{2}\) (d) \(2 \sqrt{3}\)

The angle between the tangents drawn from the point \((1,4)\) to the parabola \(y^{2}=4 x\) is (a) \((\pi / 2)\) (b) \((\pi / 3)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

The vertices of the hyperbola \(9 x^{2}-16 y^{2}-36 x+96 y-252=0\) are (a) \((6,3),(-6,3)\) (b) \((-6,3),(-6,-3)\) (c) \((6,-3),(2,-3)\) (d) \((6,3),(-2,3)\)
See all solutions

Recommended explanations on English Textbooks

Language and Social Groups

Read Explanation

Phonology

Read Explanation

Cues and Conventions

Read Explanation

Argumentative Essay

Read Explanation

Essay Prompts

Read Explanation

Single Paragraph Essay

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free