Topic 4.6 Conic Sections: identify and analyze parabolas, ellipses, circles and hyperbolas from their equations, and describe their key features.

What this topic is asking

The College Board (Topic 4.6) wants you to identify and analyze the conic sections: parabolas, ellipses, circles and hyperbolas. Each has a standard implicit equation whose structure encodes the shape, and from that equation you read the center, radius or semi-axes, vertices and orientation.

The four standard forms

The constants $(h, k)$ are the shifts that move the center or vertex away from the origin, exactly the transformations of Topic 1.11.

Telling the conics apart

This algebraic signature identifies the conic before you compute any feature, which makes the analysis systematic.

The circle as a special ellipse

A point worth stating once is that a circle is the special case of an ellipse with equal semi-axes. When $a = b$ the ellipse equation $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{a^2} = 1$ multiplies out to $(x - h)^2 + (y - k)^2 = a^2$, a circle of radius $a$. So the family runs continuously: stretch a circle unequally and it becomes an ellipse; the circle is just the symmetric member. Recognizing this keeps you from treating the circle and ellipse as unrelated and explains why the circle's parametrisation (Topic 4.4) generalizes to the ellipse by using different radii in the cosine and sine terms.

A related point concerns the parabola, which stands apart from the other three. The circle, ellipse and hyperbola all have two squared terms, so they are symmetric about their center; the parabola has only one squared term and so has a single vertex rather than a center, and it opens in one direction only. In $y = a(x - h)^2 + k$ the vertex is $(h, k)$ and the sign of $a$ tells you whether it opens up ($a > 0$) or down ($a < 0$); the form $x = a(y - k)^2 + h$ opens right or left instead. This is the same vertex-form analysis you met for quadratic polynomials in Unit 1, now seen as one of the four conic sections. Keeping the parabola's single-squared-term signature in mind is the fastest way to separate it from the ellipse and hyperbola at a glance.

Try this

Q1. What conic is $x^2 + y^2 = 49$, and what is its radius? [1 point]

• Cue. A circle centered at the origin with radius $\sqrt{49} = 7$.

Q2. Which way does $\frac{y^2}{4} - \frac{x^2}{9} = 1$ open? [1 point]

• Cue. The $y$-term is positive, so the hyperbola opens up and down.