Problem 16 - Analytical Geometry: 2D and 3D

Question

\(P\) is a variable point on the line \(y=b\), prove that the polar of \(P\) with respect to the parabola \(y^{2}=4 a x\) is a fixed directrix.

Step-by-Step Solution

Answer

Question: Prove that the polar of a variable point P on the line y = b with respect to the parabola y² = 4ax is a fixed directrix. Answer: To prove that the polar of a variable point P on the line y = b with respect to the parabola y² = 4ax is a fixed directrix, we first find the equation of the tangent to the parabola. Then, we determine the reflection of P with respect to the tangent. Next, we find the equation of the line passing through P and its reflection. Finally, we show that this line is a fixed directrix of the parabola.

Step 1: Define the polar of a point

Given the definition, the polar of a point P with respect to a curve is a straight line whose points are equidistant from P and its reflections on tangents to the curve at each of the points of contact. We will later generate these points that are equidistant to P and the tangents drawn to the parabola.

Step 2: Find the equation of the tangent to the parabola

The tangent of a parabola y^2 = 4ax at some point T(t) can be given by the equation: \(y(t) = 2a\left(1 + t^2\right), x(t) = at^2\). So, the equation of the tangent is: \(y = t\left(x - at^2\right)+ 2at\).

Step 3: Determine the reflection of P with respect to the tangent

Let the point P be on the line y = b, so its coordinates will be \((x_p, b)\). We'll find the reflection of point P with respect to the tangent line by using the formula for reflection. Let the reflection of P with respect to the tangent be Q. We can write that as: Q = P - 2(P - T(t)), where T(t) represents the point of tangency on the parabola. Therefore, the coordinates of the point Q are: \(x_q=x_p-2\left(x_p - at^2\right)\) \(y_q=b - 2\left(b - 2at\right)\)

Step 4: Find the equation of the line passing through P and Q

Now, the polar of P with respect to the parabola must pass through both P and Q. We can find the equation of this line by using the point-slope form. The slope of the line PQ is \(\frac{y_q - b}{x_q - x_p}\). Substituting the coordinates of P and Q, we get: \(\frac{y-b}{x-x_p}=\frac{b-2\left(b-2at\right)-b}{x_p-2\left(x_p - at^2\right)-x_p}\) Solving this equation for y, we get: \(y = \frac{b\left(x - x_p\right)}{x_p- at^2}+ b\)

Step 5: Prove that the polar of P is a fixed directrix

The polar of P with respect to the parabola is the line y = b, which is a horizontal line. We see that this line does not depend on any variables. Therefore, it is a fixed directrix of the parabola y^2 = 4ax.

Key Concepts

Parabola

In analytical geometry, a parabola is a type of conic section, formed by intersecting a right circular cone with a plane parallel to its side. Its defining feature is a symmetrical, open shape that can be represented by the equation \(y^2 = 4ax\), where \(a\) is a constant that determines the width and orientation of the parabola. Parabolas are significant in many fields such as physics and engineering due to their reflective properties.

Here, the parabola given is \(y^2 = 4ax\), indicating it is symmetrical around the x-axis. The vertex of this parabola is at the origin, (0,0), and it opens to the right because \(a\) is positive. Parabolas are uniquely defined by their vertex and focus, and they possess an axis of symmetry. Understanding these features allows us to explore further properties like tangents and directrices, which are crucial in understanding curves.

Polar of a Point

The concept of a polar of a point in relation to a curve is an interesting aspect of analytical geometry. When you have a curve, such as a parabola, the polar of a point \(P\) outside the curve is the locus of points that are equidistant from \(P\) and its reflections on tangents to the curve. This involves reflecting the point \(P\) across various tangent lines of the parabola, generating a straight line called the polar. This line is fixed for a specific curve and serves an important role in geometric constructions.

In this exercise, \(P\) is a variable point on \(y = b\), a horizontal line. By examining how \(P\) relates to the parabola \(y^2 = 4ax\), we can determine that the polar of \(P\) with respect to this parabola is constant, thus demonstrating a fixed directrix.

Directrix

A directrix is a crucial element in the geometry of a parabola. It is a fixed line used in defining and constructing the parabola, similar to the focus of the parabola. Together, the focus and the directrix define each point on the parabola as being equidistant from the focus and the directrix.

For the parabola \(y^2 = 4ax\), its directrix is a vertical line located at \(x = -a\). This is integral to the parabola's architecture. In this specific problem, the line \(y = b\) acts as the polar of \(P\) with respect to the parabola. By establishing that this line always relates back to the parabola, we show how it forms a fixed directrix, maintaining the symmetrical properties and constraints of the parabola.

Tangent to a Curve

The tangent of a curve, like a parabola, is a line that just touches the curve at a point without cutting across it. It indicates the slope or instantaneous rate of change at that point.

For the parabola \(y^2 = 4ax\), the equation of the tangent at any point \(T(t)\) on the curve can be given by \(y = t(x - at^2) + 2at\), which derives from the general properties of curves. This line plays a significant role in reflecting and determining points for the polar of a curve.

By reflecting point \(P\) from this tangent, we engage in geometric construction that contributes to determining the position and orientation of the polar, hence affecting the directrix and overall shape perception of the parabola.