Try the fastest way to create flashcards
Question

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola.

x=5y2+25y+60x=5 y^2+25 y+60

Solution

Verified
Answered 2 years ago
Answered 2 years ago
Step 1
1 of 2

Rewrite in standard form.

x=5y+25+60x=5[y+5y+(b2)2]+605(b2)2x=5[y+5y+(52)2]+605(52)2x=5[y+5y+(2.5)2]+605(2.5)2x=5(y+5y+6.25)+605(6.25)x=5(y+2.5)2+28.75\begin{aligned} x&=5y+25+60\\ x&=5\left[y+5y+\left(\frac{b}{2}\right)^2\right]+60-5\left(\frac{b}{2}\right)^2\\ x&=5\left[y+5y+\left(\frac{5}{2}\right)^2\right]+60-5\left(\frac{5}{2}\right)^2\\ x&=5\left[y+5y+\left(2.5\right)^2\right]+60-5\left(2.5\right)^2\\ x&=5\left(y+5y+6.25\right)+60-5(6.25)\\ x&=5(y+2.5)^2+28.75 \end{aligned}

Thus, the standard form is

x=5(y+2.5)2+28.75\begin{aligned} x&=5(y+2.5)^2+28.75 \end{aligned}

Therefore, vertex=(28.75,2.5)\text{vertex}=(28.75,-2.5) axis of symmetry=28.75\text{axis of symmetry}=28.75

focus=(h+14a,k)=(28.75+14(5),2.5)=(28.75+120,2.5)=(57620,2.5)\begin{aligned}\text{focus}&=\left (h+\dfrac{1}{4a},k\right)\\ &=\left (28.75+\frac{1}{4(5)},-2.5\right)\\ &=\left (28.75+\frac{1}{20},-2.5\right)\\ &=\left (\frac{576}{20},-2.5\right)\\ \end{aligned}

focus=(28.8,3)\text{focus}=\left (28.8,-3\right)\\

directrix:x=h14a=28.7514(5)=28.75(120)=575120=57420\begin{aligned}\text{directrix}:x&=h-\dfrac{1}{4a}\\ &=28.75-\dfrac{1}{4(5)}\\ &=28.75-\left(\frac{1}{20}\right)\\ &=\frac{575-1}{20}\\ &=\frac{574}{20} \end{aligned}

directrix=28.7\text{directrix}=28.7 a>0=opens to the righta>0= \text {opens to the right}

lenght of latus rectum=1a unit=15\begin{aligned}\text{lenght of latus rectum}&=\dfrac{1}{a}\text{ unit}\\&=\dfrac{1}{5}\\ \end{aligned}

latus rectum=15 unit\text{latus rectum}=\frac{1}{5}\text{ unit}

Create a free account to view solutions

Create a free account to view solutions

Recommended textbook solutions

Algebra 2, California Edition 1st Edition by Beatrice Moore Harris, Carter, Casey, Cuevas, Day, Hayek, Holliday, Marks

Algebra 2, California Edition

1st EditionISBN: 9780078659805Beatrice Moore Harris, Carter, Casey, Cuevas, Day, Hayek, Holliday, Marks
8,110 solutions
enVision Algebra 2 1st Edition by Al Cuoco

enVision Algebra 2

1st EditionISBN: 9780328931590 (1 more)Al Cuoco
3,573 solutions
Big Ideas Math Algebra 2: A Common Core Curriculum 1st Edition by Boswell, Larson

Big Ideas Math Algebra 2: A Common Core Curriculum

1st EditionISBN: 9781608408405 (1 more)Boswell, Larson
5,067 solutions
Big Ideas Math: Algebra 2 A Common Core Curriculum 1st Edition by Laurie Boswell, Ron Larson

Big Ideas Math: Algebra 2 A Common Core Curriculum

1st EditionISBN: 9781642088052Laurie Boswell, Ron Larson
5,067 solutions

More related questions

1/4

1/7