Question

Show that the set of points equidistant from a circle and a line not passing through the circle is a parabola. Assume the circle, line, and parabola lie in the same plane.

Short answer

Question: Prove that the locus of points equidistant from a given circle and a line (assuming they don't intersect) is a parabola. Short answer: By defining a focus (F) and directrix (L) and choosing a point Q on the set of equidistant points (P), we show that the distance from Q to the focus F is equal to the distance from Q to the directrix D, using the properties of similar triangles and the Pythagorean theorem. This proves that the set P of equidistant points is indeed a parabola.

Step-by-step solution

Define the circle, line, and point

Let's define the circle with center O and radius r, and a line L that doesn't pass through the circle. Let P be the set of points equidistant from the circle and line L.

Define the focus and directrix

Choose a point F on the circle closest to line L. Draw a line segment from F perpendicular to L and call the intersection point on L as D. The focus (F) and directrix (L) form part of the definition of a parabola.

Choose a point on the parabola

Choose a point Q on the set P of equidistant points from the circle and line L.

Find the distance from Q to the circle and line L

Since Q is equidistant from the circle and line L, we can find the distances: d(Q, circle) = d(Q, line L).

Find the point on the circle closest to Q

Connect Q to the center O of the circle to form the segment OQ. Let M be the point on the circle closest to Q such that OM intersect the circle at G. Now, the shortest distance between a point and a circle is the perpendicular from the point to the circle's radius, so QM is perpendicular to OM.

Find the distance from Q to F

Connect the points F and Q, forming the segment FQ. We need to show that d(Q, F) = d(Q, D).

Use the properties of triangles to simplify the problem

Since QM is perpendicular to OM and FD is perpendicular to DL, triangles QMG and FDG are similar triangles (by AA similarity). Then, we know the lengths of their sides are proportional. From the proportionality between the sides of the triangles, we have: \[\frac{QM}{FD} = \frac{OG}{DG}\] We also know that OG = OM - MG = OM - r, and DG = OM - OD = OM - FL. Substitute these values in the proportion: \[\frac{QM}{FD} = \frac{OM - r}{OM - FL}\]

Find QM and FL

To find QM and FL, notice that triangle QFD is also a right triangle with right angle at D. From the Pythagorean theorem, we have: \[QF^2 = QD^2 - FD^2\] Since QD = FD (because Q is equidistant from the circle and the line), QF^2 = FD^2, or: \[QF = FD\] Now, we can rewrite our proportion from step 7: \[\frac{QM}{QF} = \frac{OM - r}{OM - FL}\] Since both sides of the equation equal to 1: \[QM = QF\]

Conclusion

We've shown that the distance from any point Q on the parabola to the focus F is equal to the distance from Q to the directrix D, meaning that our set P of equidistant points is indeed a parabola.

Key concepts

Geometric Definition of Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point, known as the focus, and a fixed line, called the directrix.

The geometric beauty of a parabola lies in this simple yet powerful definition. It's not just any arbitrary curve; it’s a precise collection of points that each share a common property—equal distance to the focus and directrix. Imagining a point moving across the plane, it draws the graceful U-shaped curve when keeping this equilibrium of distances. This balance is the very nature of a parabola and why traffic parabolic mirrors give drivers a wide field of vision, focusing light into intense beams in satellite dishes or creating the perfect arc of a basketball shot.

Focus and Directrix of Parabola

Every parabola has two defining features: the focus and the directrix.

The focus is the point towards which the parabola shapes itself. In a physical sense, like in the reflective properties of a dish antenna, the signals all reflect and 'focus' at this point. The directrix, on the other hand, is a line that the parabola never touches but remains equally distant to along its curve.

• The point that provides this perfect symmetry, the focus, is not randomly set but calculated based on the distance from the directrix.
• Similarly, the directrix is a strategic line that works in tandem with the focus to define the curve of the parabola.

Understanding the intricate dance between these two elements is critical in mapping the parabola’s exact path through space.

Similar Triangles in Geometry

Triangles are a fundamental building block in geometry, and similar triangles are pairs of triangles that have the same shape but possibly different sizes. This means they have equal corresponding angles and proportional corresponding sides.

Through similarity, we can solve many geometric problems by creating proportions–ratios that describe the relationship between the lengths of their sides. For instance, in the case of our parabola problem, recognizing that two triangles are similar allowed us to set up a proportion that led to proving the equidistance required for a parabola. Besides, similar triangles are everywhere around us—bridges, towers, and even in art—and these geometric relationships are powerful tools in engineering and design.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, this can be written as: \[ a^2 + b^2 = c^2 \]where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other two sides. This theorem not only helps in calculating distances but is also an indispensable tool in various fields such as physics, engineering, and even in computer science for tasks that involve calculating distances between points on a screen. Understanding and applying the Pythagorean theorem was critical in the process of confirming the equidistant relationship in our parabola exercise, showcasing the theory's incredible utility and timeless nature.