Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![### Problem Statement
Given the focus and directrix, find the value of \(\frac{1}{4p}\).
- **Focus**: \((4, 5)\)
- **Directrix**: \(y = 11\)
### Options
- a) \(-\frac{1}{12}\)
- b) \(\frac{1}{24}\)
- c) \(\frac{1}{12}\)
- d) \(-\frac{1}{24}\)
### Explanation
To solve this problem, typically, the parabola in question is vertical since the directrix is a horizontal line \(y = 11\). The vertex is midway between the focus \((4, 5)\) and the directrix \(y = 11\).
The distance between the focus and directrix is \(11 - 5 = 6\). Therefore, the vertex is at \(y = 8\) (average of 5 and 11), and the complete relationship in terms of \(p\) (distance from vertex to focus or directrix) can be found.
Then, compute \(\frac{1}{4p}\) based on this information, selecting from the given options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3aa69a92-6482-48de-ba12-49603e3a7822%2Fcd287046-d446-43f5-a59c-7b5618543540%2Fizp323v_processed.png&w=3840&q=75)
![](/static/compass_v2/shared-icons/check-mark.png)
Focus: (4, 5)
Directrix: y = 11
As we know, and
Here, (h, k) is center.
Therefore,
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