The question you've posed involves a hyperbola defined by the equation xy = c², and it entails analyzing the properties of tangents and normals at a point on the hyperbola. To tackle this, let’s break down the components step-by-step and derive the relationship between the areas of the triangles formed.
Understanding the Hyperbola and Points
First, consider a point P on the hyperbola, which can be represented as (x₀, y₀). Since it lies on the hyperbola, we have the relationship:
xy = c² implies x₀y₀ = c².
Finding the Tangent and Normal Lines
The slope of the hyperbola at point P can be derived through implicit differentiation. The equation of the tangent line at point P can be expressed as:
y - y₀ = m_T(x - x₀),
where m_T is the slope of the tangent line. For the hyperbola xy = c², the slope at point P can be calculated to be:
m_T = -y₀/x₀.
Thus, the equation of the tangent line becomes:
y - y₀ = -\frac{y₀}{x₀}(x - x₀).
Finding the Intersections with the Axes
To find the intersection T on the x-axis (where y = 0), set y = 0 in the tangent equation:
0 - y₀ = -\frac{y₀}{x₀}(x - x₀),
solving this gives the x-coordinate of T. Similarly, for T₁ on the y-axis (where x = 0):
y - y₀ = -\frac{y₀}{x₀}(0 - x₀).
Defining the Normal Line
The normal line at point P has a slope that is the negative reciprocal of the tangent’s slope. Therefore:
m_N = x₀/y₀.
The equation for the normal line is:
y - y₀ = \frac{x₀}{y₀}(x - x₀).
Finding Intersections for the Normal Line
Similar to the tangent, we can find where the normal intersects the axes. For point N on the x-axis:
0 - y₀ = \frac{x₀}{y₀}(x - x₀),
and for point N₁ on the y-axis:
y - y₀ = \frac{x₀}{y₀}(0 - x₀).
Calculating the Areas of the Triangles
Now, let’s compute the areas of triangles formed by points P, N, and T, as well as P, N', and T'. The area of a triangle formed by points (x₁, y₁), (x₂, y₂), and (x₃, y₃) can be computed using the formula:
Area = \frac{1}{2} \left| x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) \right|.
For triangle PNT, substituting the coordinates will yield the area Δ₁. Similarly, you can find Δ' for triangle PN'T'.
Deriving the Final Relationship
Now, the relationship you’re interested in is:
\(\frac{1}{\Delta} + \frac{1}{\Delta'}\). This can be derived from the areas calculated above. Through careful manipulation of the expressions for Δ₁ and Δ', it's possible to show that:
\(\frac{1}{\Delta} + \frac{1}{\Delta'} = \frac{2}{c²}\).
This result encapsulates the relationship between the areas of the triangles formed by the tangent and normal lines at point P on the hyperbola. The exercise beautifully illustrates the interplay between geometry and algebra in the study of conic sections!
