Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Conic Sections

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #12 : Circles

Find the equation of the circle if it is centered at $\small (6,4)$ and has a radius of $\small 5$ units.

Possible Answers:

$\small (x-6) + (y-4) = 5$

$\small (x-6)^2 + (y-4)^2 = 25$

$\small (x+6)^2 + (y+4)^2 = 25$

Correct answer:

$\small (x-6)^2 + (y-4)^2 = 25$

Explanation:

The equation of a circle centered at $\small (h,\,k) \,$ with radius units in standard form is

$\small (x-h)^2+(y-k)^2=r^2$

For the circle ceentered at $\small \small (6,4)$ with radius $\small 5$ units has the equation

$\small (x-6)^2 + (y-4)^2 = 5^2$

$\small (x-6)^2 + (y-4)^2 = 25$

Report an Error

Example Question #13 : Circles

(3,6) is a point on a circle whose center is at (-2,-4) . What is the standard equation of this circle?

Possible Answers:

(x-3)^2+(y-6)^2=125

(x+2)^2+(y+4)^2=125

(x+3)^2+(y+6)^2=125

None of the other answers.

(x-2)^2+(y-4)^2=125

Correct answer:

(x+2)^2+(y+4)^2=125

Explanation:

The standard form of the equation of a circle is

(x-h)^2+(y-k)^2=r^2

where the center of the circle resides at the point (h,k) .

Given the center of a circle and a point on the rim of the circle, one can use the distance formula to find the radius.

$d=r=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}=\sqrt{(3+2)^2+(6+4)^2}=\sqrt{125}$

Now plug in the point for the center and radius into the standard equation of a circle:

$(x+2)^2+(y+4)^2=(\sqrt{125})^2\rightarrow \mathbf{(x+2)^2+(y+4)^2=125}$

Report an Error

Example Question #14 : Circles

Rewrite the following in standard form:

x^2+y^2-6x+12y+20=0

Possible Answers:

(x+3)^2+(y-6)^2=4^2

(x-3)^2+(y+6)^2=5^2

(x-6)^2+(y+3)^2=5^2

(x-6)^2+(y+3)^2=4^2

(x+3)^2+(y-6)^2=5^2

Correct answer:

(x-3)^2+(y+6)^2=5^2

Explanation:

To solve for general form, we want to format the given equation as (x-h)^2+(y-k)^2=r^2

$x^2+y^2-6x+12y+20=0\\ x^2-6x+\qquad+y^2+12y+\qquad=-20\\ (x^2-6x+9)+(y^2+12y+36)=-20+9+36\\ (x-3)^2+(y+6)^2=25$

(x-3)^2+(y+6)^2=5^2

Report an Error

Example Question #15 : Circles

Rewrite the following equation in standard form:

x^2+y^2+4x+8y-29=0

Possible Answers:

(x+2)^2+(y-4)^2=5^2

(x-2)^2+(y-4)^2=7^2

(x-2)^2+(y+4)^2=6^2

(x-2)^2+(y-4)^2=5^2

(x+2)^2+(y+4)^2=7^2

Correct answer:

(x+2)^2+(y+4)^2=7^2

Explanation:

To rewrite in standard form, we must format the equation as (x-h)^2+(y-k)^2=r^2

$x^2+y^2+4x+8y-29=0\\ x^2+4x+\qquad+y^2+8y+\qquad=29\\ (x^2+4x+4)+(y^2+8y+16)=29+4+16\\ (x+2)^2+(y+4)^2=49\\ (x+2)^2+(y+4)^2=7^2$

Report an Error

Example Question #16 : Determine The Equation Of A Circle In Standard Form

Rewrite the given equation in standard form:

x^2+y^2+14x-6y+54=0

Possible Answers:

(x-3)^2+(y+7)^2=3^2

(x-7)^2+(y+3)^2=2^2

(x+7)^2+(y-9)^2=8^2

(x+7)^2+(y-3)^2=2^2

(x-7)^2+(y+3)^2=1^2

Correct answer:

(x+7)^2+(y-3)^2=2^2

Explanation:

To rewrite in standard form, we must follow the format (x-h)^2+(y-k)^2=r^2

$x^2+y^2+14x-6y+54=0\\ x^2+14x+\qquad+y^2-6y+\qquad=-54\\ (x^2+14x+49)+(y^2-6y+9)=-54+49+9\\ (x+7)^2+(y-3)^2=4\\ (x+7)^2+(y-3)^2=2^2$

Report an Error

Example Question #21 : Circles

Write the equation for x^2 +y^2 -6x+8y +4 =0 in standard form

Possible Answers:

(x-3)^2 +(y+4)^2 = 21

(x-3)^2 + (y+4)^2 = 4

(x+3)^2 +(y+4)^2 = 29

(x-6)^2 + (y-8)^2 = 4

(x-3)^2 + (y-4)^2 =21

Correct answer:

(x-3)^2 +(y+4)^2 = 21

Explanation:

To determine the standard-form equation, we'll have to complete the square for both x and y. It will be really helpful to re-group our terms to do that:

$x^2 -6x \enspace \enspace \enspace + y^2 +8y \enspace \enspace \enspace = -4$

Adding 9 will complete the square for x, since $(\frac{1}{2}(-6))^2 = (-3)^2 = 9$

Adding 16 will complete the square for y, since $(\frac{1}{2}\cdot8)^2 = 4^2 = 16$

x^2 -6x +9 + y^2 +8y +16 = -4 +9 + 16

Now we just need to simplify. Re-write the left side as two binomials squared, and add the numbers on the right side:

(x-3)^2 + (y+4)^2 = 21

Report an Error

Example Question #21 : Circles

If the center of the circle is (5,-9) and the radius is 6, what is the equation of the circle in standard form?

Possible Answers:

(x-5)^2 - (y+9)^2 =6

(x-5)^2 + (y+9)^2 =6

(x-5)^2 + (y+9)^2 =36

(x+5)^2 - (y-9)^2 =36

(x+5)^2 + (y-9)^2 =36

Correct answer:

(x-5)^2 + (y+9)^2 =36

Explanation:

Write the equation of the standard form of the circle.

(x-a)^2 + (y-b)^2 = r^2

where (a,b) is the center of the circle and is the radius.

Substitute the center and the radius into the equation.

(x-5)^2 + (y-(-9))^2 = 6^2

Reduce this equation and leave this in standard form.

(x-5)^2 + (y+9)^2 =36

Report an Error

Example Question #21 : Circles

Determine the equation of a circle whose center is at (1,1) and radius is .

Possible Answers:

(x+1)^2+(y+1)^2=1

(x-1)^2+(y-1)^2=2

(x-1)^2+(y-1)^2=1

(x+1)^2+(y+1)^2=1

Correct answer:

(x-1)^2+(y-1)^2=1

Explanation:

To solve, simply use the formula for a circle as given below.

(x-h)^2+(y-k)^2=r^2

Thus, our answer is:

(x-1)^2+(y-1)^2=1

Report an Error

Example Question #22 : Circles

Determine the equation for a circle in standard form, centered at (3,-4), with radius 2.

Possible Answers:

(x-3)^2+(y+4)^2=2

(x+3)^2+(y-4)^2=4

(x+3)^2+(y-4)^2=2

(x-3)^2+(y+4)^2=4

(x-3)^2-(y+4)^2=4

Correct answer:

(x-3)^2+(y+4)^2=4

Explanation:

Recall that the standard from for the equation of a circle is

(x-h)^2+(y-k)^2=r^2

where (h, k) is the center, and r is the radius. We are given the center (3, -4) and radius 2. Therefore, h = 3, k = -4, and r = 2. Plugging these vaules into the equation gives us

(x-3)^2+(y-(-4))^2=2^2

(x-3)^2+(y+4)^2=4

Report an Error

Example Question #25 : Circles

A circle centered at (6,1) passes through (11,13). Write an equation for the circle in standard form.

Possible Answers:

(x+6)^2+(y+1)^2=144

(x-6)^2+(y-1)^2=144

(x-6)^2+(y-1)^2=121

(x+6)^2+(y+1)^2=169

(x-6)^2+(y-1)^2=169

Correct answer:

(x-6)^2+(y-1)^2=169

Explanation:

Recall the equation of a circle in standard form:

(x-h)^2+(y-k)^2=r^2 , where (h, k) is the center and r is the radius.

In this problem, we are given the center, but no radius. We must use the other piece of information to find the radius. The second point given is a point on the circle. The definition of a radius is the distance between the center and any point on the circle. Therefore, the radius is equal to the distance between (6,1) and (11,13). Using the distance formula,

$d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$

$d=\sqrt{(11-6)^2+(13-1)^2}$

$d=\sqrt{(5)^2+(12)^2}$

$d=\sqrt{25+144}$

d=13

Therefore, the radius is 13. Plugging all the information into the standard form of a circle gives us

(x-6)^2+(y-1)^2=13^2

(x-6)^2+(y-1)^2=169

Report an Error

View Pre-Calculus Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Pre-Calculus Tutors

Emmanuel
Certified Tutor

Baylor University, Bachelor of Science, Biology, General.

View Pre-Calculus Tutors

Denys
Certified Tutor

Lviv National University Ukraine, Bachelors, Biology/Biophysics. University of Szeged Hungary, PHD, Computational Biophysics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Algebra Tutors in Los Angeles, SAT Tutors in Boston, ACT Tutors in Dallas Fort Worth, GMAT Tutors in Seattle, Physics Tutors in Phoenix, SAT Tutors in Phoenix, Chemistry Tutors in San Diego, ACT Tutors in Miami, Biology Tutors in New York City, ACT Tutors in Phoenix

Popular Courses & Classes

ACT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Los Angeles, ACT Courses & Classes in Washington DC, LSAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Seattle, GMAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Atlanta, Spanish Courses & Classes in Chicago

Popular Test Prep

GRE Test Prep in Washington DC, LSAT Test Prep in Denver, SAT Test Prep in Houston, ISEE Test Prep in New York City, ISEE Test Prep in Denver, ACT Test Prep in Atlanta, ISEE Test Prep in Boston, ACT Test Prep in Philadelphia, ACT Test Prep in Washington DC, LSAT Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Conic Sections

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #12 : Circles

Example Question #13 : Circles

Example Question #14 : Circles

Example Question #15 : Circles

Example Question #16 : Determine The Equation Of A Circle In Standard Form

Example Question #21 : Circles

Example Question #21 : Circles

Example Question #21 : Circles

Example Question #22 : Circles

Example Question #25 : Circles

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: