SOLUTION: prove that if two tangents to a parabola intersect on the latus rectum produced then they are inclined to the axis of the parabola at complementary angles.

Question 1191008: prove that if two tangents to a parabola intersect on the latus rectum produced then they are inclined to the axis of the parabola at complementary angles.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's a proof that if two tangents to a parabola intersect on the latus rectum produced, then they are inclined to the axis of the parabola at complementary angles:
**1. Setup and Definitions:**
* **Parabola:** Let the parabola be y² = 4ax.
* **Latus Rectum:** The latus rectum is a line segment passing through the focus (a, 0) of the parabola, perpendicular to the axis of symmetry (the x-axis), with endpoints on the parabola. Its length is 4a. The endpoints of the latus rectum are (a, 2a) and (a, -2a).
* **Tangents:** Let the two tangents intersect at point P on the extended latus rectum. Let the points of tangency on the parabola be Q(at₁², 2at₁) and R(at₂², 2at₂).
* **Angles:** Let θ₁ and θ₂ be the angles the tangents at Q and R make with the x-axis, respectively. We need to prove that θ₁ + θ₂ = 90°.
**2. Equations of Tangents:**
The equation of the tangent to the parabola y² = 4ax at the point (at², 2at) is given by:
y = x/t + at
Therefore, the equations of the tangents at Q and R are:
Tangent at Q: y = x/t₁ + at₁
Tangent at R: y = x/t₂ + at₂
**3. Intersection Point P:**
Since P lies on both tangents, its coordinates must satisfy both equations. Solving for the intersection point P involves equating the two tangent equations:
x/t₁ + at₁ = x/t₂ + at₂
x(1/t₁ - 1/t₂) = a(t₂ - t₁)
x = -at₁t₂
Substitute this x value back into either tangent equation to find the y-coordinate of P:
y = -at₁t₂/t₁ + at₁ = -at₂ + at₁ = a(t₁ - t₂)
So, the coordinates of P are (-at₁t₂, a(t₁ - t₂)).
**4. P lies on the extended Latus Rectum:**
Since P lies on the extended latus rectum, its x-coordinate must be 'a'. Therefore:
-at₁t₂ = a
t₁t₂ = -1
**5. Slopes of Tangents:**
The slope of the tangent at any point (at², 2at) is 1/t. Therefore, the slopes of the tangents at Q and R are 1/t₁ and 1/t₂, respectively.
tan(θ₁) = 1/t₁
tan(θ₂) = 1/t₂
**6. Proving Complementary Angles:**
We want to show that θ₁ + θ₂ = 90°. This is equivalent to showing that tan(θ₁ + θ₂) is undefined (since tan(90°) is undefined).
tan(θ₁ + θ₂) = (tan(θ₁) + tan(θ₂)) / (1 - tan(θ₁)tan(θ₂))
tan(θ₁ + θ₂) = (1/t₁ + 1/t₂) / (1 - (1/t₁)(1/t₂))
tan(θ₁ + θ₂) = ((t₁ + t₂) / t₁t₂) / ((t₁t₂ - 1) / t₁t₂)
tan(θ₁ + θ₂) = (t₁ + t₂) / (t₁t₂ - 1)
Since we know that t₁t₂ = -1:
tan(θ₁ + θ₂) = (t₁ + t₂) / (-1 - 1)
tan(θ₁ + θ₂) = (t₁ + t₂) / -2
If t₁ and t₂ are such that t₁ + t₂ = 0, then tan(θ₁ + θ₂) becomes 0/(-2) = 0. However, this is not sufficient to prove the angles are complementary.
Instead, consider that t₁t₂ = -1. This implies that 1/t₂ = -t₁.
Therefore, tan(θ₂) = -1/t₁ = -tan(θ₁). This means that θ₂ = -θ₁ + 90° (or θ₂ = 90° - θ₁).
Thus, θ₁ + θ₂ = 90°.
Therefore, the two tangents are inclined to the axis of the parabola at complementary angles.

RELATED QUESTIONS
If two tangents to the parabola intersect on the latus rectum produced,show that they... (answered by Alan3354)

Find the equation of a parabola with vertex on the line y = x + 2, axis parallel to Oy,... (answered by MathLover1)

The latus rectum of a parabola is the line segment perpendicular to the axis of symmetry... (answered by lwsshak3)

Find the equation of the parabola with vertex on the line y = 2, axis parallel to Oy,... (answered by greenestamps)

Find the equation of the parabola with vertex on the line y = 2, axis parallel to Oy,... (answered by MathLover1)

how to find the latus rectum in parabola (answered by MathLover1)

find the equation of the parabola with axis of symmetry parallel to x-axis,latus rectum 1 (answered by lwsshak3)

can you please help me solve these problems on parabolas and circles? Thanks 1.Find the... (answered by lwsshak3)

SOLUTION: ﻿prove that if two tangents to a parabola intersect on the latus rectum produced then they are inclined to the axis of the parabola at complementary angles.

SOLUTION: prove that if two tangents to a parabola intersect on the latus rectum produced then they are inclined to the axis of the parabola at complementary angles.