Read conics from equations and geometric definitions
Complete squares and distinguish ellipses, hyperbolas and parabolas.
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Read x²+y²−4x+6y−12=0 as a geometric curve. Complete both squares: (x−2)²+(y+3)²=25. The points lie at distance 5 from (2,−3), so the curve is a circle. The signs inside the squares identify the centre; they are not the centre coordinates unchanged.
Predict
Predict how replacing the equal squared coefficients by x²/16+y²/4=1 changes the circle. The horizontal reach becomes 4 and vertical reach 2, producing an ellipse. Replacing the plus between squared terms by a minus creates a different family with two separate branches.
Experiment
Compare unshifted standard forms on equal-scale axes.
Curve
Equation
Geometric landmarks
Circle
x²+y²=9
Radius 3
Ellipse
x²/16+y²/4=1
Semiaxes 4 and 2
Hyperbola
x²/9−y²/4=1
Vertices (±3,0)
Parabola
y²=8x
Vertex (0,0), opens right
For the ellipse, plot (4cos t,2sin t) at quarter-turn parameters and verify by substitution. For the hyperbola, solve y²=4(x²/9−1) to see why |x| must be at least 3. Its asymptotes are y=±2x/3, suggesting the directions of its distant branches.
Understand
A circle fixes distance from a centre. An ellipse fixes the sum of distances to two foci. For horizontal semiaxes a>b>0, its foci are at (±c,0), where c²=a²−b². A hyperbola fixes the absolute difference of distances to two foci; in x²/a²−y²/b²=1, c²=a²+b². These focal relationships differ, so copying the ellipse subtraction into a hyperbola calculation is incorrect.
A parabola consists of points equidistant from a focus and a directrix line. With focus (p,0) and directrix x=−p, for p>0, the distance equation is √((x−p)²+y²)=|x+p|. Squaring gives y²=4px. Both original distances are nonnegative, so squaring is reversible here. The vertex is (0,0), midway between focus and directrix.
Shifts replace x by x−h and y by y−k, moving a centre or vertex to (h,k). For a shifted ellipse, x=h+a cos t and y=k+b sin t provides a parametrization. The full major-axis length is 2a, not a, when a is the larger semiaxis.
This introduction treats axis-aligned, nondegenerate standard forms. General second-degree equations can include an xy term, rotations, empty real loci or degenerate pairs of lines. Their complete classification and detailed focal proofs require further analytic-geometry work; not every equation containing squares describes one of the pictured nondegenerate curves.
Apply
Complete squares carefully, adding the compensating constants to the correct side. Distinguish semiaxis from full axis and centre from focus. In the parabola derivation, explain why the two quantities being squared are nonnegative rather than assuming every squaring step preserves solutions.
Continue
This sixteen-lesson unit supplies trigonometric and representation foundations. It does not by itself certify complete AP Precalculus coverage: broader modeling assessments, polar-rate work, matrix functions and full analytic geometry remain separate requirements. Continue with cumulative unfamiliar problems and later calculus of parametric and polar curves.
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Practice and reasoning
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For x²+y²−6x+4y−12=0, find the circle radius.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For x²/25+y²/9=1, what is the full major-axis length?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For y²=12x, what is the x-coordinate of the focus?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which equation describes a hyperbola rather than an ellipse?
Give a parametrization of(x−2)²/16+(y+1)²/4=1 tracing it once, and identify its centre and semiaxes.
Structure your argument and preserve revisions
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Derive the parabola with focus(2,0) and directrix x=−2 from equality of distances. Explain why squaring introduces no extraneous curve here.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
