Toni filled her car with gas before going on a road trip. The car uses 1.75 gallons of gas for every hour driven, and after driv
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Answer:
The equation after translation is x² + y² + 16x - 8y + 73 = 0 ⇒ answer (a)
Step-by-step explanation:
* Lets study the type of the equation:
∵ Ax² + Bxy + Cy² + Dx + Ey + F = 0 ⇒ general form of conic equation
- If D and E = zero
∴ The center of the graph is the origin point (0 , 0)
- If B = 0
∴ The equation is that of a circle
* Lets study our equation:
x² + y² = 7 ⇒ x² + y² - 7 =0
∵ B = 0 , D = 0 , E = 0
∴ It is the equation of a circle with center origin
- The equation of the circle with center origin in standard form is:
x² + y² = r²
∴ x² + y² = 7 is the equation of a circle withe center (0 , 0)
and its radius = √7
* We have two translation one horizontally and the other vertically
- Horizontal: x-coordinate moves right (+ve value) or left (-ve value)
- Vertical: y-coordinate moves up (+ve value) down (-ve value)
∵ The point of translation is (-8 , 4)
∵ x = -8 (-ve value) , y = 4 (+ve value)
∴ The circle moves 8 units to the left and 4 units up
* now lets change the x- coordinate and the y-coordinate
of the center (0 , 0)
∴ x-coordinate of the center will be -8
∵ y-coordinate of the center will be 4
* That means the center of the circle will be at point (-8 , 4)
- the standard form of the equation of the circle with center (h , k) is
(x - h)² + (y - k)² = r²
∵ h = -8 and y = 4
∴ The equation is: (x - -8)² + (y - 4)² = 7
∴ (x + 8)² + (y - 4)² = 7
* lets change the equation to the general form by open the brackets
∴ x² + 16x + 64 + y² - 8y + 16 - 7 = 0
* Lets collect the like terms
∴ x² + y² + 16x - 8y + 73 = 0
∴ The equation after translation is x² + y² + 16x - 8y + 73 = 0
* Look at the graph the blue circle is after translation
