Grade 12 · Implicit circle tangents · 82 of 600
Implicit circle tangents · Circle radius r=3; Final angle (degrees)=180
Use angle θ=108°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary. Givens: Circle radius r=3; Final angle (degrees)=180.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
The circle is x²+y²=9, with radius 3. Its polar angle is measured in degrees, and the animation ends at 180°. Use angle θ=108°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary.
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A starting point
First use x=r cos θ and y=r sin θ. Differentiate the circle equation to obtain 2x+2y(dy/dx)=0.
Work through the reasoning
Step 1
Identify the model and target
The circle is x²+y²=9, with radius 3. Its polar angle is measured in degrees, and the animation ends at 180°. Use angle θ=108°. The governing relation is x²+y²=r²; dy/dx=−x/y when y≠0. First use x=r cos θ and y=r sin θ. Differentiate the circle equation to obtain 2x+2y(dy/dx)=0.
Step 2
Substitute and calculate
The point is (3 cos(108°), 3 sin(108°))=(-0.927051,2.85317), with squared radius 9. Implicit differentiation gives dy/dx=−(-0.927051)/(2.85317)=0.32492.
Step 3
Check the mathematical meaning
The radius vector is normal to the tangent direction (−y,x), since their dot product is −xy+yx=0. Here y=2.85317 is nonzero, so −x/y=0.32492 is a valid slope. x coordinate: -0.927; y coordinate: 2.853; Radius: 3; Slope −x/y: 0.325. Decimal values are rounded, so use unrounded intermediate values.
The answer
The point is (3 cos(108°), 3 sin(108°))=(-0.927051,2.85317), with squared radius 9. Implicit differentiation gives dy/dx=−(-0.927051)/(2.85317)=0.32492. The radius vector is normal to the tangent direction (−y,x), since their dot product is −xy+yx=0. Here y=2.85317 is nonzero, so −x/y=0.32492 is a valid slope. Animation check: x coordinate: -0.927; y coordinate: 2.853; Radius: 3; Slope −x/y: 0.325. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
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The mathematical idea
An implicit equation describes both halves of a circle without choosing one as a function of x. Differentiating relates the tangent slope to the current coordinates. At y=0 the tangent is vertical, so the slope formula has no finite value even though the circle itself remains smooth. The circle is x²+y²=9, with radius 3. Its polar angle is measured in degrees, and the animation ends at 180°. Use angle θ=108°. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
x²+y²=r²; dy/dx=−x/y when y≠0
03 · Reflect and transfer
Explain what changes and why.
At which two points does the slope formula fail, and why does the geometric tangent still exist there?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.