Coordinate Geometry: Straight Lines, Circles, Parabola, Ellipse and Hyperbola in Two Dimensions

To find the center, first divide the equation of the circle by 4 so the leading coefficients are 1:

Compare this with the general form of the circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$:
$2g = -2 \implies g = -1$
$2f = 3 \implies f = \frac{3}{2}$

The center of the circle is given by $(-g, -f)$:

For a standard vertical parabola opening downwards:

Comparing $x^2 = -9y$ with the standard form:

The length of the latus rectum of any parabola is defined as the geometric distance $|4a|$.
Since length is always a positive scalar, the length of the latus rectum is 9.

Analysis of eccentricity across conic sections:
By definition, eccentricity $e$ is the constant ratio of the distance of a point on the conic from the focus to its perpendicular distance from the directrix ($SP / PM = e$).

• Correct: For a parabola, the distance to the focus equals the distance to the directrix ($SP = PM$), so eccentricity $e = 1$.
• Incorrect: $e = 0$ corresponds to a circle.
• Incorrect: $e \lt 1$ corresponds to an ellipse.
• Incorrect: $e \gt 1$ corresponds to a hyperbola.

The standard upward-opening parabola is $x^2 = 4ay$.
Comparing $x^2 = \frac{25}{2}y$ with $x^2 = 4ay$ gives:
$4a = \frac{25}{2} \implies a = \frac{25}{8}$.
The directrix of $x^2 = 4ay$ is horizontal and located below the vertex at $y = -a$.