Problem 32 - Analytical Geometry: 2D and 3D

Question

Two tangents drawn from a point to the parabola make angles \(\theta_{1}\) and \(\theta_{2}\) with the \(x\) -axis. Show that the locus of their point of intersection if \(\tan ^{2} \theta_{1}+\) \(\tan ^{2} \theta_{2}=c\) is \(y^{2}-c x^{2}=2 a x\).

Step-by-Step Solution

Answer

Question: Find the locus of the intersection of two tangents to the parabola \(y^2 = 4ax\) that make angles θ₁ and θ₂ with the x-axis. Answer: The locus equation is \(y^2 - cx^2 = 2ax\).

Step 1: Write down the equation of a tangent to a parabola

The equation of a tangent to the parabola \(y^2 = 4ax\) at the point \((xt, yt)\) can be written as: \(y\cdot y_{t}=2a(x+x_{t})\)

Step 2: Compute the tangent slopes in terms of the given angles

To find the tangent slopes, we know that \(m_{\text {tangent 1}}=\tan \left(\theta_{1}\right)\) and \(m_{\text {tangent 2}}=\tan \left(\theta_{2}\right)\).

Step 3: Calculate the point of intersection (H, K) using the slope formulas

Let's denote the point of intersection of the tangents as \((H, K)\). The equation of the tangents passing through \((H, K)\) can be written as: $$ y - K = \tan (\theta_{1})(x - H) \\ y - K = \tan (\theta_{2})(x - H) $$ Now we have a system of two equations and two unknowns (H and K).

Step 4: Determine the relationship between the point of intersection and the given angles

To relate the coordinates of the intersection point to the angles, we can eliminate one of the unknowns (either H or K) from the system of equations above. We'll eliminate K by subtracting the two equations: $$ \tan(\theta_{2})(x - H) - \tan(\theta_{1})(x - H) = y - \tan(\theta_{1}) (x - H) - y $$ Simplifying the equation, we get: $$ \left(\tan(\theta_{1}) + \tan(\theta_{2})\right)(x - H) = 0 $$ Since the sum of the tangents, \(\tan(\theta_{1}) + \tan(\theta_{2})\), is nonzero, the only way for this equation to hold is when \(x = H\). Now, we can plug in x = H back into the equation of the tangent to get: $$ y\cdot y_{t} = 2a(x + x_{t}) $$ Replacing x with H and using the equation \(y^2 = 4ax\), we obtain: $$ K^2 -cH^2= 2aH $$ This is the required locus equation: $$ y^2 -cx^2= 2ax $$

Key Concepts

Analytical Geometry

Analytical geometry, also known as coordinate geometry, is a branch of mathematics that uses algebraic equations to describe the properties of geometric shapes. The cornerstone of analytical geometry is the coordinate system, typically the Cartesian coordinate system, where points are defined by an ordered pair of numbers referred to as coordinates.

The Cartesian plane, constituted by the x and y axes, allows us to express geometric entities, such as lines, circles, and parabolas, as mathematical relationships. For example, a parabola in the Cartesian plane can be represented by the general equation \( y^2 = 4ax \), where \( a \) is a constant that determines the parabola’s width and orientation.

Through the principles of analytical geometry, we can analyze and solve problems involving distances, midpoints, gradients, and curves by using algebraic methods. Therefore, students tackling exercises related to geometrical shapes and their properties often use algebraic strategies to identify key characteristics like the focus, directrix, or the locus of points satisfying certain conditions.

Parabola Properties

A parabola is a symmetrical, open plane curve that is formed by the intersection of a cone with a plane parallel to its side. Several key properties of a parabola make it an interesting subject of study in geometry:

• The vertex is the highest or lowest point on the parabola, depending on its orientation.
• The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• The focus and directrix are elements that help define the parabola. Each point on the parabola is equidistant from the focus and the directrix.
• The latus rectum is a line segment through the focus and perpendicular to the axis of symmetry, with both endpoints lying on the parabola.

When discussing the properties of parabolas in analytic geometry, it's essential to understand how these features translate into algebraic form, particularly when working with equations that represent a parabola's orientation, shape, and position in the coordinate plane.

Tangent Lines

In geometric terms, a tangent line to a curve at a given point is the straight line that just touches the curve at that point. This line has the same slope as the curve at that point of contact and represents an instantaneous direction of the curve.

The concept of tangencies is pivotal when studying curves like parabolas since it helps analyze their geometry. In the case of a parabola \( y^2 = 4ax \), we can determine the equation of a tangent at a particular point \( (xt, yt) \). The standard form for this tangent is \( y \times yt = 2a(x + xt) \), which can be derived from the parabola’s equation.

What's fascinating is how we can explore the characteristics of tangents, such as their slope represented by \( m = \tan(\theta) \), where \( \theta \) is the angle the tangent makes with the x-axis. By delving into these relationships between a parabola and its tangent lines, students can gain a deeper understanding of the curve's properties and the applications of these in problem-solving.