give the standard form of the circle whose center is (-10,3) and radius is 7. Show your solution.
Answers
EXPLANATION.
Standard equation of circle,
Whose center of circle = (-10,3).
Radius of the circle = 7.
As we know that,
General equation of circle,
⇒ x² + y² + 2gx + 2fy + c = 0.
We can write as,
⇒ (x - h)² + (y - k)² = (r)².
⇒ Center = (-10,3) = (h, k).
⇒ r = 7.
Put the values in the equation, we get.
⇒ (x - (-10))² + (y - 3)² = (7)².
⇒ (x + 10)² + (y - 3)² = (7)².
As we know that,
Formula of :
⇒ (x + y)² = x² + y² + 2xy.
Using this formula in equation, we get.
⇒ (x² + 100 + 20x) + (y² + 9 - 6y) = 49.
⇒ x² + 100 + 20x + y² + 9 - 6y - 49 = 0.
⇒ x² + y² + 20x - 6y + 60 = 0.
MORE INFORMATION.
The parametric equation of a circle.
(1) = The parametric equation of a circle x² + y² = r². are x = r cosθ , y = r sinθ.
(2) = The parametric equations of the circle (x - h)² + (y - k)² = r² are x = h + r cosθ , y = k + r sinθ.
(3) = Parametric equations of the circle x² + y² + 2gx + 2fy + c = 0 are x = - g + √g² + f² - c cosθ , y = - f + √g² + f² - c sinθ.
Explaination :-
Standard equation of circle,
Whose center of circle = (-10,3).
Radius of the circle = 7.
As we know that,
General equation of circle,
→ x² + y² + 2gx + 2fy + c = 0.
We can write as,
→ (xh)² + (y-k)² = (r)².
→ Center = (-10,3)= (h, k).
⇒ r = 7.
Put the values in the equation, we get.
→ (x - (-10))² + (y - 3)² = (7)².
→ (x+10)² + (y - 3)² = (7)².
As we know that,
Formula of:
⇒ (x + y)² = x² + y² + 2xy.
Using this formula in equation, we get.
(x² + 100 + 20x) + (y² + 9 - 6y) = 49.
→ x² + y² + 20x - 6y + 60 = 0.
→ x² + 100 + 20x + y² +9-6y - 49 = 0.
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