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Algebra 2 Flashcards: Derive The Equation Of A Circle

Study Derive The Equation Of A Circle in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra 2

Derive The Equation Of A Circle

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QUESTION
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What is the general form after expanding (x−2)2+(y+3)2=16(x-2)^2+(y+3)^2=16?

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ANSWER

x2+y2−4x+6y−3=0x^2+y^2-4x+6y-3=0. Expand and rearrange to general form.

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What this deck covers

This deck focuses on Derive The Equation Of A Circle, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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Flashcard 1: What is the general form after expanding (x−2)2+(y+3)2=16(x-2)^2+(y+3)^2=16?

Answer: x2+y2−4x+6y−3=0x^2+y^2-4x+6y-3=0. Expand and rearrange to general form.

Flashcard 2: What is the radius of a circle with center (1,−2)(1,-2) passing through (4,2)(4,2)?

Answer: 55. Distance formula: (4−1)2+(2−(−2))2=5\sqrt{(4-1)^2+(2-(-2))^2}=5.

Flashcard 3: What is the center of the circle x2+y2−2x+14y+33=0x^2+y^2-2x+14y+33=0?

Answer: (1,−7)(1,-7). From standard form, center is (1,−7)(1,-7).

Flashcard 4: What is the equation of the circle centered at (−2,1)(-2,1) passing through (1,5)(1,5)?

Answer: (x+2)2+(y−1)2=25(x+2)^2+(y-1)^2=25. Distance from (−2,1)(-2,1) to (1,5)(1,5) is r=5r=5.

Flashcard 5: What is the equation of the circle with center (0,7)(0,7) and radius 22?

Answer: x2+(y−7)2=4x^2+(y-7)^2=4. Substitute center (h,k)=(0,7)(h,k)=(0,7) and radius r=2r=2.

Flashcard 6: What is the equation of the circle centered at (0,0)(0,0) passing through (−3,4)(-3,4)?

Answer: x2+y2=25x^2+y^2=25. Distance from origin to (−3,4)(-3,4) is r=5r=5.

Flashcard 7: What is the center of the circle (x−3)2+(y+5)2=16(x-3)^2+(y+5)^2=16?

Answer: (3,−5)(3,-5). From (x−3)2+(y+5)2(x-3)^2+(y+5)^2, center is (3,−5)(3,-5).

Flashcard 8: What is the completed-square form of x2+Dxx^2+Dx?

Answer: (x+D2)2−(D2)2\left(x+\frac{D}{2}\right)^2-\left(\frac{D}{2}\right)^2. Perfect square trinomial minus the added constant.

Flashcard 9: What is the general form after expanding (x+5)2+(y−1)2=36(x+5)^2+(y-1)^2=36?

Answer: x2+y2+10x−2y−10=0x^2+y^2+10x-2y-10=0. Expand and rearrange to general form.

Flashcard 10: What is the standard form of x2+y2−2x+14y+33=0x^2+y^2-2x+14y+33=0?

Answer: (x−1)2+(y+7)2=17(x-1)^2+(y+7)^2=17. Complete square with D=−2D=-2, E=14E=14, F=33F=33.

Flashcard 11: What is the completed-square form of y2+Eyy^2+Ey?

Answer: (y+E2)2−(E2)2\left(y+\frac{E}{2}\right)^2-\left(\frac{E}{2}\right)^2. Perfect square trinomial for yy terms.

Flashcard 12: What is the equation of the circle with center (−4,6)(-4,6) and radius 33?

Answer: (x+4)2+(y−6)2=9(x+4)^2+(y-6)^2=9. Substitute center (h,k)=(−4,6)(h,k)=(-4,6) and radius r=3r=3.

Flashcard 13: What is the general form after expanding x2+(y−4)2=20x^2+(y-4)^2=20?

Answer: x2+y2−8y−4=0x^2+y^2-8y-4=0. Expand and rearrange to general form.

Flashcard 14: What is the center and radius of (x+1)2+(y−2)2=0(x+1)^2+(y-2)^2=0?

Answer: Center (−1,2)(-1,2); radius 00. Degenerate circle: single point at (−1,2)(-1,2).

Flashcard 15: What is the completed-square expression equivalent to y2+8yy^2+8y?

Answer: (y+4)2−16(y+4)^2-16. Complete square: (y+4)2=y2+8y+16(y+4)^2=y^2+8y+16.

Flashcard 16: What is the center and radius of x2+y2+2x+6y+1=0x^2+y^2+2x+6y+1=0?

Answer: Center (−1,−3)(-1,-3); radius 33. Complete square: D=2D=2, E=6E=6, F=1F=1.

Flashcard 17: What is the radius in terms of diameter endpoints (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2)?

Answer: r=12(x2−x1)2+(y2−y1)2r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. Radius is half the distance between endpoints.

Flashcard 18: What midpoint formula gives the center of a circle from diameter endpoints (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2)?

Answer: (x1+x22,y1+y22)\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). Midpoint formula gives center from endpoints.

Flashcard 19: What is the completed-square expression equivalent to x2−12xx^2-12x?

Answer: (x−6)2−36(x-6)^2-36. Complete square: (x−6)2=x2−12x+36(x-6)^2=x^2-12x+36.

Flashcard 20: What is the radius of the circle with diameter endpoints (2,−1)(2,-1) and (8,5)(8,5)?

Answer: 323\sqrt{2}. Half the distance between diameter endpoints.

Flashcard 21: What is the center of x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0 after completing the square?

Answer: (−D2,−E2)\left(-\frac{D}{2},-\frac{E}{2}\right). Center coordinates are negatives of linear coefficients.

Flashcard 22: What is the standard form equation of a circle with center (h,k)(h,k) and radius rr?

Answer: (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2. Standard form shows center (h,k)(h,k) and radius rr explicitly.

Flashcard 23: What is the standard form of x2+y2−8x+2y−8=0x^2+y^2-8x+2y-8=0?

Answer: (x−4)2+(y+1)2=25(x-4)^2+(y+1)^2=25. Complete square with D=−8D=-8, E=2E=2, F=−8F=-8.

Flashcard 24: What is the equation of the circle with diameter endpoints (−2,4)(-2,4) and (4,−2)(4,-2)?

Answer: (x−1)2+(y−1)2=18(x-1)^2+(y-1)^2=18. Midpoint (1,1)(1,1), radius half diagonal length.

Flashcard 25: What is the center and radius of x2+y2+4x−10y+13=0x^2+y^2+4x-10y+13=0?

Answer: Center (−2,5)(-2,5); radius 44. Complete square: D=4D=4, E=−10E=-10, F=13F=13.

Flashcard 26: What is the Pythagorean distance formula between (x,y)(x,y) and (h,k)(h,k)?

Answer: d=(x−h)2+(y−k)2d=\sqrt{(x-h)^2+(y-k)^2}. Distance formula derived from Pythagorean theorem.

Flashcard 27: Identify the value added to complete the square for x2−14xx^2-14x.

Answer: 4949. Take half of coefficient: (−14/2)2=49(-14/2)^2=49.

Flashcard 28: What is the standard form of x2+y2+10x+12y+9=0x^2+y^2+10x+12y+9=0?

Answer: (x+5)2+(y+6)2=52(x+5)^2+(y+6)^2=52. Complete square with D=10D=10, E=12E=12, F=9F=9.

Flashcard 29: What must be true about the coefficients of x2x^2 and y2y^2 for a circle in general form?

Answer: They are equal and nonzero. Both coefficients must be 1 for a true circle.

Flashcard 30: What is the equation of the circle with diameter endpoints (0,0)(0,0) and (6,8)(6,8)?

Answer: (x−3)2+(y−4)2=25(x-3)^2+(y-4)^2=25. Midpoint (3,4)(3,4), radius half diagonal length.

Flashcard 31: What is the standard form of x2+y2+6x−4y−12=0x^2+y^2+6x-4y-12=0?

Answer: (x+3)2+(y−2)2=25(x+3)^2+(y-2)^2=25. Complete square with D=6D=6, E=−4E=-4, F=−12F=-12.

Flashcard 32: What is the general form of a circle equation after expanding standard form?

Answer: x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0. Result of expanding standard form and collecting terms.

Flashcard 33: What is the radius of the circle x2+y2+10x+12y+9=0x^2+y^2+10x+12y+9=0?

Answer: 52\sqrt{52}. From standard form, r2=52r^2=52, so r=52r=\sqrt{52}.

Flashcard 34: What is the equation of a circle with center (h,k)(h,k) passing through (x1,y1)(x_1,y_1)?

Answer: (x−h)2+(y−k)2=(x1−h)2+(y1−k)2(x-h)^2+(y-k)^2=(x_1-h)^2+(y_1-k)^2. Distance from center to point equals the radius.

Flashcard 35: What is the equation of a circle centered at the origin with radius rr?

Answer: x2+y2=r2x^2+y^2=r^2. When center is at origin, h=0h=0 and k=0k=0.

Flashcard 36: What is the center and radius of the circle (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2?

Answer: Center (h,k)(h,k); radius rr. Read center and radius directly from standard form.

Flashcard 37: What is the radius of the circle x2+y2=49x^2+y^2=49?

Answer: 77. Since r2=49r^2=49, take square root to get r=7r=7.

Flashcard 38: What does it mean if a circle equation gives r2=0r^2=0 after completing the square?

Answer: The circle is a single point (a degenerate circle). Circle degenerates to single point when r2=0r^2=0.

Flashcard 39: What is the center and radius of x2+y2−12x−2y+37=0x^2+y^2-12x-2y+37=0?

Answer: Center (6,1)(6,1); radius 00. Complete square: r2=36+1−37=0r^2=36+1-37=0.

Flashcard 40: What is the equation of the circle with center (2,−1)(2,-1) and radius 55?

Answer: (x−2)2+(y+1)2=25(x-2)^2+(y+1)^2=25. Substitute center (h,k)=(2,−1)(h,k)=(2,-1) and radius r=5r=5.

Flashcard 41: What is r2r^2 for x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0 in terms of D,E,FD,E,F?

Answer: (D2)2+(E2)2−F\left(\frac{D}{2}\right)^2+\left(\frac{E}{2}\right)^2-F. Sum of completing square constants minus FF.

Flashcard 42: What is r2r^2 for a circle centered at (h,k)(h,k) passing through (x1,y1)(x_1,y_1)?

Answer: r2=(x1−h)2+(y1−k)2r^2=(x_1-h)^2+(y_1-k)^2. Radius squared equals distance squared to point.

Flashcard 43: Identify the value added to complete the square for y2+9yy^2+9y.

Answer: 814\frac{81}{4}. Take half of coefficient: (9/2)2=81/4(9/2)^2=81/4.

Flashcard 44: What is the equation of the circle in standard form: x2+y2=9x^2+y^2=9?

Answer: (x−0)2+(y−0)2=9(x-0)^2+(y-0)^2=9. Already in standard form with center (0,0)(0,0).

Flashcard 45: What is the equation of the circle with center (−3,0)(-3,0) and radius 1010?

Answer: (x+3)2+y2=100(x+3)^2+y^2=100. Substitute center (h,k)=(−3,0)(h,k)=(-3,0) and radius r=10r=10.

Flashcard 46: What is the first step to complete the square in x2+Dxx^2+Dx?

Answer: Add and subtract (D2)2\left(\frac{D}{2}\right)^2. Add (D2)2\left(\frac{D}{2}\right)^2 to create perfect square trinomial.

Flashcard 47: What is the radius of the circle (x−3)2+(y+5)2=16(x-3)^2+(y+5)^2=16?

Answer: 44. Since r2=16r^2=16, take square root to get r=4r=4.

Flashcard 48: What is the center of the circle with diameter endpoints (2,−1)(2,-1) and (8,5)(8,5)?

Answer: (5,2)(5,2). Midpoint formula for diameter endpoints.

Flashcard 49: What does it mean if completing the square gives r2<0r^2<0?

Answer: No real circle exists (no real points satisfy it). Negative r2r^2 means no real solutions exist.

Flashcard 50: What equation results from setting distance from (x,y)(x,y) to (h,k)(h,k) equal to rr?

Answer: (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2. Circle equation comes from distance formula equal to rr.

Flashcard 51: What is the center and radius of x2+y2−6x+8y−11=0x^2+y^2-6x+8y-11=0?

Answer: Center (3,−4)(3,-4); radius 66. Complete square: D=−6D=-6, E=8E=8, F=−11F=-11.

Flashcard 52: What is the equation of the circle centered at (3,4)(3,4) passing through (6,8)(6,8)?

Answer: (x−3)2+(y−4)2=25(x-3)^2+(y-4)^2=25. Distance from (3,4)(3,4) to (6,8)(6,8) is r=5r=5.

Flashcard 53: What is the center and radius of (x−7)2+(y+3)2=81(x-7)^2+(y+3)^2=81?

Answer: Center (7,−3)(7,-3); radius 99. Read center and radius from standard form.