12 multiple-choice questions, progressively harder.
Complete the squares in x2+y2−8x−10y+32=0x^2+y^2-8x-10y+32=0x2+y2−8x−10y+32=0. What are the center and radius?
Solution
Correct answer: C
Move 32 and add 16 and 25 to complete both squares.
The center is (4,5) and the positive radius is 3. The other choices reverse the center signs or read an incorrect radius.
A diameter has endpoints (−7,−2)(-7,-2)(−7,−2) and (1,4)(1,4)(1,4). Which circle equation fits it?
Correct answer: A
The midpoint of the diameter is (-3,1), so it is the center. From that center to either endpoint the coordinate gaps have magnitudes 4 and 3.
The squared radius is 25. Using 100 would square the full diameter instead, while 5 is the radius rather than its square.
Find every common point of the circle and line.
Correct answer: B
After substituting, the vertical gap is twice the horizontal gap.
The first coordinates are negative 1 and 3; the line then gives heights negative 6 and 2. Each makes the circle left side 20. The other proposed pairs omit a root or contain points failing one equation.
Which description fits the common points of these graphs?
Correct answer: D
Complete both squares.
The graph contains only (-1,3), which satisfies the line x equals negative 1. There is one common point, but no nonzero radius and no positive-radius circle to which the tangent criterion applies.
A circle has center (−2,1)(-2,1)(−2,1) and radius 5. Which line is tangent at T=(1,5)T=(1,5)T=(1,5)?
The point T is on the circle because its coordinate gaps are 3 and 4, whose squared sum is 25. The radius slope is 4/3, so the tangent slope is negative 3/4.
The equation with right side 17 misses T. The other two pass through T but have slopes 4/3 and negative 4/3, neither perpendicular to the radius.
Classify P=(7,1)P=(7,1)P=(7,1) relative to the graph x2+y2−6x+4y−7=0x^2+y^2-6x+4y-7=0x2+y2−6x+4y−7=0.
Completing the squares gives
The squared center distance of P is
The right side 20 is positive, so a real circle exists; 25 exceeds 20, putting P outside.
The line cuts a chord from the circle below. What is the chord length?
Substitute the line height.
The two first coordinates are 1 and 7. The chord is horizontal, so its length is
Three is only half the chord, the diameter has length twice the square root of 13, and 9 is a squared half-length.
For which real values of b is the line tangent to this circle?
A horizontal tangent has its contact directly above or below the center (-1,3), at one radius distance. The radius is 3.
The two heights are 0 and 6. At either height the only first coordinate is negative 1, and the vertical contact radius is perpendicular to the horizontal line. Height 3 cuts through the center; height negative 3 misses the circle.
What is the radius of 2x2+2y2−20x−8y+48=02x^2+2y^2-20x-8y+48=02x2+2y2−20x−8y+48=0?
Divide the whole equation by 2 before completing the squares.
The positive radius is the square root of 5. None of the other proposed radii has square 5.
The circle x2+y2−2x+4y+c=0x^2+y^2-2x+4y+c=0x2+y2−2x+4y+c=0 passes through (4,2)(4,2)(4,2). Find c.
Substitute the known boundary point into the original equation.
Thus c is negative 20. Completing the squares would then give squared radius 25, confirming that a positive-radius circle results. The other constants fail the boundary-point substitution.
The line passes through T at (0,−3)(0,-3)(0,−3) on the circle below. Which description is correct?
The center is (-4,-3), so the given horizontal line passes through the center as well as T. Substitution gives
There are two intersections, (0,-3) and (-8,-3). A tangent at T would be vertical, perpendicular to the horizontal contact radius.
For this circle and line, what does substitution show?
The line fixes the vertical gap from the center at 3. Substitute it.
A real square cannot equal negative 6, so there is no intersection. Each other option uses a different squared equation from the actual substitution.
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