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Given an ellipse $x^2/a^2+y^2/b^2=1,$ where $a\not=b,$ find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.

First I let $P(c,d)$ outside the ellipse,then assume the linear equation: $y=(x-c)+d;$ second, according to the problem, the line must touch the ellipse, so substitute $y$ for the equation of ellipse to get the intersection, then let $D(\text{discriminant})=0.$ After all of this, I have no idea how to continue.

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  • $\begingroup$ Why don't you determine the point pairs on the ellipse first and then determine the intersection of the tangents. Express the ellipse as a y = function and then take the derivative and solve for dy/dx = c and 1/c. $\endgroup$
    – Phil H
    Commented May 16, 2018 at 14:29
  • $\begingroup$ Seems like there’s something missing from your equation of a line. $\endgroup$
    – amd
    Commented May 16, 2018 at 19:59
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    $\begingroup$ where was first case a) $ m_1m_2= +1$ answered? $\endgroup$
    – Narasimham
    Commented May 17, 2018 at 2:51
  • $\begingroup$ @Narasimham Fair point. This answer can be adapted for that case. $\endgroup$
    – amd
    Commented May 18, 2018 at 1:09
  • $\begingroup$ But how adapted? There should be a solution preferably with a sketch. $\endgroup$
    – Narasimham
    Commented May 18, 2018 at 9:27

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