Associating a focal length to an angle for a parabola

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SUMMARY

The discussion focuses on determining the focal length 'l' of a point 'P' on a parabola defined by the equation y² = 4ax, where 'P' is determined by a line from the focus (a, 0) making an angle θ with the x-axis. The intersection point 'P' is expressed as (a - l cos(θ), l sin(θ)). The analysis reveals that the quadratic equation derived from substituting 'P' into the parabola's equation yields two solutions for 'l', which indicates the presence of two intersection points: one corresponding to the positive slope and the other to the negative slope of the line through the focus.

PREREQUISITES
  • Understanding of parabolic equations, specifically y² = 4ax.
  • Knowledge of basic trigonometry, particularly angle and slope relationships.
  • Familiarity with quadratic equations and their discriminants.
  • Concept of focal length in conic sections.
NEXT STEPS
  • Explore the properties of parabolas and their focal points in conic sections.
  • Study the derivation and implications of the quadratic formula in relation to geometric intersections.
  • Investigate the significance of the discriminant in determining the nature of solutions for quadratic equations.
  • Learn about the geometric interpretation of angles and slopes in relation to conic sections.
USEFUL FOR

Students studying conic sections, mathematicians interested in geometric properties, and educators teaching advanced algebra or calculus concepts.

rohanprabhu
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Homework Statement


If there is a line from the focus of a parabola such that it makes an angle \theta with the x-axis. It intersects the parabola at a point 'P'. I want to find the focal length of the point P as a function of \theta.

The Attempt at a Solution


The equation of the parabola is given by y^2 = 4ax. The focus is given by (a, 0).

The point where the line from the focus intersects the parabola is 'P' and the focal length of the point is given by 'l'. Then, the point 'P' is given by: (a - l \cos ({\theta}), l \sin ({\theta})).

Then, to get, i substitute in the equation for a parabola and get a quadratic equation in 'l'. The problem i face now is in explaining in how exactly i get two solutions for 'l' [discriminant is dependent on certain factors and not necessarily 0]. For a given theta, it's pretty clear that i will be getting a certain length only. Then why is it so that i can get more than two solutions. What does the second solution signify?
 
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The other solution is the where the other line at angle \theta through (a, 0), the one with negative slope, intersects the parabola.
 

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