Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Answer:
(i) Center = (-1, -2)
(ii) Radius = 5 units
General Circle Equation:
- (x - h)² + (y - k)² = r²
Where:
- (h, k) is the center points
- r denotes the radius
Rewriting the equation:
[tex]\sf x^2 + 2x + y^2+ 4y = 20[/tex]
[tex]\sf x^2 + 2x + 1^2 - 1^2 + y^2 + 4y + 2^2 - 2^2 = 20[/tex]
[tex]\sf (x + 1)^2 - 1 + (y + 2)^2 -4 = 20[/tex]
[tex]\sf (x + 1)^2 + (y + 2)^2 = 20 + 5[/tex]
[tex]\sf (x + 1)^2 + (y + 2)^2 = 25[/tex]
[tex]\sf (x -(- 1))^2 + (y -(- 2))^2 = 5^2 \quad \leftarrow \ \bf General \ Circle \ Equation[/tex]
Identify the following:
- (h, k) = (-1, -2), radius = 5 units
Answer:
Completing the square: Circles
Add the square of half the coefficients of both first degree terms (x and y) to both sides:
[tex]\begin{aligned}\implies x^2+2x+\left(\dfrac{2}{2}\right)^2+y^2+4y+\left(\dfrac{4}{2}\right)^2 & =20+\left(\dfrac{2}{2}\right)^2+\left(\dfrac{4}{2}\right)^2\\\\x^2+2x+1+y^2+4y+4 & = 20+1+4\\\\x^2+2x+1+y^2+4y+4 & = 25\end{aligned}[/tex]
Factor the two trinomials on the left side of the equation:
[tex]\begin{aligned} \implies x^2+2x+1+y^2+4y+4 & = 25\\\\ \implies (x+1)^2+(y+2)^2 & = 25 \end{aligned}[/tex]
Equation of a circle: [tex](x-a)^2+(y-b)^2=r^2[/tex]
(where (a, b) is the center and r is the radius)
Comparing constants:
[tex]\displaystyle (x-a)^2+(y-b)^2=r^2\\\\\phantom{(((((}\downarrow \phantom{(((((((((} \downarrow \phantom{(((((} \downarrow \\\\(x+1)^2+(y+2)^2=25[/tex]
Therefore:
[tex]-a=1 \implies a=-1[/tex]
[tex]-b=2 \implies b=-2[/tex]
[tex]r^2=25 \implies r=\sqrt{25}=5[/tex]
Conclusion:
- center = (-1, -2)
- radius = 5 units
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
find the equation of a line in slope intercept form, passing through (2,5) and parallel to 2x+3y=-12