Let P denote the point (8,8) on the parabola x^2=8 y, and let P Q be a focal chord. (a) Find the

step 1 number 34 here the given parabola equation is x square equals 8 y and the point p is given as 8

Let P denote the point (8,8) on the parabola x^2=8 y, and...

Let $P$ denote the point (8,8) on the parabola $x^{2}=8 y,$ and let $\overline{P Q}$ be a focal chord. (a) Find the equation of the line through the point (8,8) and the focus. (b) Find the coordinates of $Q$ (c) Find the length of $\overline{P Q}$. (d) Find the equation of the circle with this focal chord as a diameter. (e) Show that the circle determined in part (d) intersects the directrix of the parabola in only one point. Conclude from this that the directrix is tangent to the circle. Draw a sketch of the situation.

Let $P$ denote the point (8,8) on the parabola $x^{2}=8 y,$ and let $\overline{P Q}$ be a focal chord.
(a) Find the equation of the line through the point (8,8) and the focus.
(b) Find the coordinates of $Q$
(c) Find the length of $\overline{P Q}$.
(d) Find the equation of the circle with this focal chord as a diameter.
(e) Show that the circle determined in part (d) intersects the directrix of the parabola in only one point. Conclude from this that the directrix is tangent to the circle. Draw a sketch of the situation.

Key Concepts

Tangency Conditions

Tangency involves a line that touches a circle (or another curve) at exactly one point. Establishing tangency typically requires equating the distance from the center of the circle to the line with the radius of the circle, or showing that the discriminant of the system of equations equals zero. This concept is key in problems where the intersection of a circle and a line (or another curve) must be analyzed to demonstrate a unique point of contact.

Circle with a Given Diameter

A circle defined by a diameter has properties such as Thales' theorem, where any angle inscribed in a semicircle is a right angle. The construction of a circle with a focal chord as a diameter explores these properties and provides a basis for determining intersections with other figures, such as a parabola or a line, and can simplify the analysis of tangency conditions.

Focal Chord

A focal chord is a chord of a parabola that passes through the focus. It has special properties, such as a relationship between the slopes of the lines from the endpoints to the focus, and is often used to derive formulas for coordinates of the endpoints. This concept is important in understanding the intersections and symmetries present in conic sections.

Focus and Directrix

The focus is a fixed point used in the formal definition of a parabola, and the directrix is a fixed line. Every point on the parabola is equidistant from the focus and the directrix. These concepts facilitate the derivation of the parabola's equation, and are crucial when establishing relationships with other geometric figures, such as chords and circles, in problems involving reflective and tangential properties.

Parabola

A parabola is the set of points equidistant from a fixed point, called the focus, and a fixed line, called the directrix. Its standard equations can be transformed through appropriate scaling and shifting. Understanding the geometry of a parabola, including its vertex, focus, and directrix, is essential in solving problems regarding chords and tangents that involve its distinctive reflective properties.

Line Equations Through a Point

The equation of a line passing through a specific point and another point (or focus) is derived using point-slope form. In the context of a parabola, finding the line through a fixed point on the curve and the focus helps in analyzing the geometry of focal chords and understanding the intersection points with the parabola or other curves.

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