Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 15: Problem 1386

The length of the common chord of the parabolas \(\mathrm{y}^{2}=\mathrm{x}\) and \(\mathrm{x}^{2}=\mathrm{y}\) is (a) 1 (b) \(\sqrt{2}\) (c) \(4 \sqrt{2}\) (d) \(2 \sqrt{2}\)

Short Answer

Expert verified
The length of the common chord of the parabolas \(y^2=x\) and \(x^2=y\) is \(\sqrt{2}\), which corresponds to option (b).

Step by step solution

01

In order to find the intersection points, we need to solve the system of equations formed by the parabolas: \[\begin{cases} y^2 = x \\ x^2 = y \end{cases}\] We can substitute the value of x from the equation \(y^2 = x\) into the equation \(x^2 = y\) and get: \[ (y^2)^2 = y \] \[ y^4-y = 0 \] We are trying to find the common real roots of both equations for y. We can factor y out of the equation above: \[ y(y^3-1)=0 \] \[ y(y-1)(y^2+y+1)=0 \] This equation has roots y=0, y=1, and two complex roots when \(y^2+y+1=0\). Therefore, we have two intersection points with real coordinate values: \[(0,0)\] from y=0, and \[(1,1)\] from y=1. #Step 2: Calculate the length of the common chord

Now that we have the coordinates of the intersection points, we can use the distance formula to find the length of the common chord: \[ length = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the coordinates of our intersection points: \[ length = \sqrt{(1 - 0)^2 + (1 - 0)^2} \] \[ length = \sqrt{2} \] Thus, the length of the common chord of the parabolas is \(\sqrt{2}\), which corresponds to option (b).

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 curve represented by \(\mathrm{x}=3(\cos \mathrm{t}+\sin \mathrm{t})\); \(\mathrm{y}=4(\cos \mathrm{t}-\sin \mathrm{t})\) is (a) circle (b) parabola (c) ellipse (d) hyperbola

If \((\mathrm{x} / 2 \mathrm{a})+[(\mathrm{y} \sqrt{3}) / 2 \mathrm{~b}]=1\) touches the ellipse \(\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)+\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)=1\), then its eccentric angle \(\theta\) of the contact point is (a) \(0^{\circ}\) (b) \(60^{\circ}\) (c) \(45^{\circ}\) (d) \(90^{\circ}\)

The vertex of the parabola \((x-b)^{2}=4 b(y-b)\) is ........ (a) (b,0) (b) \((0, b)\) (c) \((0,0)\) (d) \((\mathrm{b}, \mathrm{b})\)

The distance from the foci of \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) on the ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 25\right)=1\) is \(\ldots \ldots \ldots\) (a) \(4-(5 / 4) \mathrm{y}_{1}\) (b) \(5-(4 / 5) \mathrm{y}_{1}\) (c) \(5-(4 / 5) \mathrm{x}_{1}\) (d) \(4-(4 / 5) \mathrm{y}_{1}\)

The area of the triangle formed by any tangent to the hyperbola \(\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)-\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)=1\) with its asymptotes is (a) \(a b\) (b) \(4 \mathrm{ab}\) (c) \(a^{2} b^{2}\) (d) \(4 \mathrm{a}^{2} \mathrm{~b}^{2}\)
See all solutions

Recommended explanations on English Textbooks

Graphology

Read Explanation

Syntax

Read Explanation

Pragmatics

Read Explanation

Email

Read Explanation

Semiotics

Read Explanation

Synthesis 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