Question

Find the equation of ellipse of the set of all points whose distances from (0,4) are (2/3) distances from the line y=9

Answer 1

Let (x,y) be any arbitrary point on the ellipse.
distance of (x,y) from (0,4), d₁= $\sqrt{(x-0)^2+(y-4)^2} =\sqrt{x^2+(y-4)^2$

distance of (x,y) from the line y-9=0, d₂ = $\frac{(y-9)}{1}=y-9$

(Note that distance of a point (X,Y) from a line ax+by+c=0 is $\frac{aX+bY+c}{ \sqrt{a^2+b^2} }$

given that d₁ = 2/3 d₂

$\sqrt{x^2+(y-4)^2}= \frac{2}{3} (y-9) \\ \\ x^2+(y-4)^2= \frac{4}{9} (y-9)^{2} \\ \\ 9(x^2+(y-4)^2)=4( y-9)^{2} \\ \\ 9 (x^{2} +y^{2} -8y+16)=4( y^{2} -18y+81)$

$9x^{2} +9y^{2} -72y+144=4y^{2} -72y+324 \\ \\9x^{2} +5y^{2}-180=0$

This is the required equation of the ellipse.