1. **Stating the problem:** We are given three equations: - $y^2=4ax$ - $x^2=4ay$ - $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ We want to understand these curves and their properties. 2. **Understanding each equation:** - The first equation $y^2=4ax$ represents a parabola opening to the right with vertex at the origin. - The second equation $x^2=4ay$ represents a parabola opening upwards with vertex at the origin. - The third equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ represents an ellipse centered at the origin with semi-major axis $a$ and semi-minor axis $b$. 3. **Important rules:** - For parabolas, the standard form is $y^2=4ax$ or $x^2=4ay$, where $a$ is the distance from the vertex to the focus. - For ellipses, the standard form is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a$ and $b$ are the semi-axes lengths. 4. **Interpreting the parameters:** - In the parabolas, $a$ controls the width and focal length. - In the ellipse, $a$ and $b$ define the shape and size. 5. **Summary:** - $y^2=4ax$ is a right-opening parabola. - $x^2=4ay$ is an upward-opening parabola. - $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is an ellipse. These are standard conic sections with parameters $a$ and $b$ defining their dimensions.