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# Find the equation of the set of all points whose distance from (0, 4) are of their distance from the line y = 9.

Answer :

Let the point be P (x, y).

According to the question,

Distance of P from (x, y)(Distance from the line y=9)

Distance between the (x,y)& (0,4) = [Distance Formula]

Perpendicular Distance (Between a point and line) = , whereas the point is and the line is expressed as ax + by + c = 0 i.e.., x(0) + y – 6 = 0 & point = (x,y)

Distance between y – 9 = 0 {x(0) +y – 9 = 0} & (x, y)

Squaring both the sides,

x^{2} + y^{2} - 8y + 16 (y^{2} - 18y + 81)

9x^{2} + 9y^{2} - 72y + 144 = 4y^{2} - 72y + 324

9x^{2} + 5y^{2} = 180

Hence, the required equation is 9x^{2} + 5y^{2} = 180

, which is an ellipse.

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