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A triangle is circumscribed to the ellipse x 2 /a ​2 + y 2 /b 2 = 0 ​and two of its vertices lie on the directories. Prove that the locus of third vertex is - b 2 y 2 = a 2 (e 2 ​ + 1) 2 (a 2 – x 2 ​) ​ ​Please solve this in a neat manner, cause I tried solving it by assuming (h, k) to be third vertex and having two tangents as the side. The vertices lie on directories and the line passing through them must also be a tangent. But this method became too complex at the end so I couldn’t solve like that.

A triangle is circumscribed to the ellipse x2/a​2 + y2/b2 = 0 ​and two of its vertices lie on the directories. Prove that the locus of third vertex is -
b2y2 = a2(e2​ + 1)2(a2 – x2​)
 
​Please solve this in a neat manner, cause I tried solving it by assuming (h, k) to be third vertex and having two tangents as the side. The vertices lie on directories and the line passing through them must also be a tangent. But this method became too complex at the end so I couldn’t solve like that.

Grade:12th pass

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