The equation of the circle with latus rectum of y^2+4 y+4...

The equation of the circle with latus rectum of $y^{2}+4 y+4 x+2=0$ as a diameter is
(a) $4\left(x^{2}+y^{2}+x+4 y\right)+1=0$
(b) $4\left(x^{2}+y^{2}+4 y\right)+1=0$
(c) $x^{2}+y^{2}+x+4 y+1=0$
(d) $x^{2}+y^{2}+3 x+4 y+1=0$

Key Concepts

Latus Rectum

The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of symmetry. Its length, typically determined by the parameter in the parabola’s equation (for example, 4a in y² = 4ax), serves as an important measure of the parabola’s width around the focus. This concept is useful when the latus rectum is used to define or construct another geometric entity, such as using it as a diameter of a circle.

Parabola

A parabola is a conic section defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Its standard equations, like y² = 4ax (or variants thereof), reveal key characteristics such as the vertex, focus, and axis of symmetry, which are important in understanding related geometric constructs.

Diameter

The diameter of a circle is a chord that passes through the center, and its length is equal to twice the radius. When a diameter is known or used in a problem, the midpoint of the diameter gives the center of the circle, and the distance between its endpoints determines the radius. This property is crucial when constructing the circle’s equation from given endpoints.

Circle (Equation and Properties)

A circle is the set of all points in a plane that are at a constant distance (the radius) from a fixed point (the center). Its equation in standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Understanding this form is essential when using given information (like a diameter) to determine the circle’s equation.

Transcript

Please give Ace some feedback