Need help on an equation of tangent lines

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SUMMARY

The discussion focuses on deriving the equation of the set of points from which two tangents to an ellipse, defined by the equation (x^2/a^2) + (y^2/b^2) = 1, are perpendicular. The user seeks assistance in determining the slopes of these tangents and understanding the condition for perpendicularity, which involves the slopes being negative reciprocals. The key challenge is to express the relationship between the point (x_0, y_0) and the tangents to the ellipse, ensuring the tangents' slopes satisfy the perpendicular condition.

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  • Understanding of ellipse equations and properties
  • Knowledge of tangent lines and their equations
  • Familiarity with the concept of slopes and perpendicular lines
  • Basic algebraic manipulation skills
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  • Learn about the conditions for perpendicular lines in coordinate geometry
  • Explore the geometric properties of ellipses and their tangents
  • Investigate the application of implicit differentiation in finding tangent equations
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kevinchhan
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Having trouble with this:

Given an ellipse (x^2/a^2) + (y^2/b^2) = , a!=b. Find the equation of the set of all points from which the two tangents to the curve are perpendicular.

I tried finding the slope of the equation then knowing that perpendicular line are the opposite reciprocal. But still clueless on how to put it together.

Thanks in advance
 
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The points in question are not on the ellipse. Suppose [itex](x_0, y_0)[/itex] is a point not on the ellipse. Then there are two lines tangent to the ellipse through [itex](x_0, y_0)[/itex]. What are there equations? What must be true if those two lines are perpendicular?
 

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