Grade 12 · Implicit circle tangents · 3 of 600
Implicit circle tangents · Circle radius r=1; Final angle (degrees)=30
Use angle θ=18°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary. Givens: Circle radius r=1; Final angle (degrees)=30.
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=1, with radius 1. Its polar angle is measured in degrees, and the animation ends at 30°. Use angle θ=18°. 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=1, with radius 1. Its polar angle is measured in degrees, and the animation ends at 30°. Use angle θ=18°. 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 (1 cos(18°), 1 sin(18°))=(0.951057,0.309017), with squared radius 1. Implicit differentiation gives dy/dx=−(0.951057)/(0.309017)=-3.077684.
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=0.309017 is nonzero, so −x/y=-3.077684 is a valid slope. x coordinate: 0.951; y coordinate: 0.309; Radius: 1; Slope −x/y: -3.078. Decimal values are rounded, so use unrounded intermediate values.
The answer
The point is (1 cos(18°), 1 sin(18°))=(0.951057,0.309017), with squared radius 1. Implicit differentiation gives dy/dx=−(0.951057)/(0.309017)=-3.077684. The radius vector is normal to the tangent direction (−y,x), since their dot product is −xy+yx=0. Here y=0.309017 is nonzero, so −x/y=-3.077684 is a valid slope. Animation check: x coordinate: 0.951; y coordinate: 0.309; Radius: 1; Slope −x/y: -3.078. 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.
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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=1, with radius 1. Its polar angle is measured in degrees, and the animation ends at 30°. Use angle θ=18°. 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.