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Zorluk: KolayEquations and Graphs of Circles

On a coordinate grid, a circle is drawn such that its center is at the point (4,−2)(4, -2). If the circle is tangent to the yy-axis, what is the equation of this circle?

  1. A
    (x+4)2+(y−2)2=16(x + 4)^2 + (y - 2)^2 = 16
  2. B
    (x−4)2+(y+2)2=4(x - 4)^2 + (y + 2)^2 = 4
  3. (x−4)2+(y+2)2=16(x - 4)^2 + (y + 2)^2 = 16Cevap
  4. D
    (x+4)2+(y−2)2=4(x + 4)^2 + (y - 2)^2 = 4
  5. E
    (x−2)2+(y+4)2=16(x - 2)^2 + (y + 4)^2 = 16

Cevap

(x−4)2+(y+2)2=16(x - 4)^2 + (y + 2)^2 = 16
The correct equation is (x−4)2+(y+2)2=16(x - 4)^2 + (y + 2)^2 = 16. A circle with center (h,k)(h, k) and radius rr has the standard equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. Substituting (4,−2)(4, -2) for the center gives (x−4)2+(y+2)2=r2(x - 4)^2 + (y + 2)^2 = r^2. Since the circle is tangent to the yy-axis, the radius is the horizontal distance from the center to the line x=0x = 0, which is 44 units. Squaring the radius gives r2=16r^2 = 16.

Adım Adım Çözüm

1
Recall the standard form of the equation of a circle.
The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius.
This sets up the formula needed to write the circle's equation.
2
Substitute the center coordinates (h,k)=(4,−2)(h, k) = (4, -2) into the formula.
(x−4)2+(y−(−2))2=r2(x - 4)^2 + (y - (-2))^2 = r^2, which simplifies to (x−4)2+(y+2)2=r2(x - 4)^2 + (y + 2)^2 = r^2.
To represent the translation of the circle's center from the origin.
3
Determine the radius of the circle using the given tangency condition.
Since the circle is tangent to the yy-axis, the radius is the horizontal distance from the center (4,−2)(4, -2) to the yy-axis (the line x=0x = 0). This distance is ∣4−0∣=4|4 - 0| = 4 units, so r=4r = 4.
To find the radius from the geometric properties of the circle.
4
Square the radius and write the complete equation.
r2=42=16r^2 = 4^2 = 16. The completed equation is (x−4)2+(y+2)2=16(x - 4)^2 + (y + 2)^2 = 16.
To complete the standard equation form.

Anahtar Kavram

Standard form of a circle's equation and determining its radius from a tangency condition
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Konular

Equations and Graphs of Circles

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