Question

Equation of the ellipse whose focus is $ (6,7) $ , directrix is $ x+y+2=0 $ and $ e=\dfrac{1}{\sqrt{3}} $ , is:
A. $ 5{{x}^{2}}+2xy+5{{y}^{2}}-76x-88y+506=0 $
B. $ 5{{x}^{2}}-2xy+5{{y}^{2}}-76x-88y+506=0 $
C. $ 5{{x}^{2}}-2xy+5{{y}^{2}}+76x+88y-506=0 $
D. None of the above.

Answer 1

Hint:Each of the two lines parallel to the minor axis, and at a distance of $
d=\dfrac{{{a}^{2}}}{c}=\dfrac{a}{e} $ from it, is called a directrix of the ellipse.

For an arbitrary point P of the ellipse, the quotient of the distance to one focus and to the
corresponding directrix is equal to the eccentricity e.

If the focus is $ F=\left( {{f}_{1}},{{f}_{2}} \right) $ , the directrix $ ux+vy+w=0 $ and the eccentricity $
e $ , one obtains the equation:
$ {{\left( x-{{f}_{1}} \right)}^{2}}+{{\left( y-{{f}_{2}} \right)}^{2}}={{e}^{2}}\dfrac{{{\left( ux+vy+w
\right)}^{2}}}{{{u}^{2}}+{{v}^{2}}} $ .

Complete step by step solution:
We know that if the focus is $ F=\left( {{f}_{1}},{{f}_{2}} \right) $ , the directrix $ ux+vy+w=0 $ and the eccentricity $ e $ , then the equation of the ellipse is: $ {{\left( x-{{f}_{1}}
\right)}^{2}}+{{\left( y-{{f}_{2}}
\right)}^{2}}={{e}^{2}}\dfrac{{{\left( ux+vy+w \right)}^{2}}}{{{u}^{2}}+{{v}^{2}}} $ .

In the given question, focus is $ F=(6,\ 7) $ , directrix is $ x+y+2=0 $ and eccentricity is $
e=\dfrac{1}{\sqrt{3}} $ .
Therefore, $ {{f}_{1}}=6,\ {{f}_{2}}=7,\ u=1,\ v=1,\ w=2,\ e=\dfrac{1}{\sqrt{3}} $ and the equation of the ellipse is:

$ {{\left( x-6 \right)}^{2}}+{{\left( y-7 \right)}^{2}}={{\left( \dfrac{1}{\sqrt{3}} \right)}^{2}}\dfrac{{{\left(
1x+1y+2 \right)}^{2}}}{{{1}^{2}}+{{1}^{2}}} $

Expanding the terms using $ {{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}} $ and $
{{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2(ab+bc+ca) $ , we get:
⇒ $ {{x}^{2}}-12x+36+{{y}^{2}}-14y+49=\left( \dfrac{1}{3}
\right)\dfrac{{{x}^{2}}+2xy+4x+{{y}^{2}}+4y+4}{2} $

Multiplying both sides by 6, we get:
⇒ $ 6{{x}^{2}}-72x+216+6{{y}^{2}}-84y+294={{x}^{2}}+2xy+4x+{{y}^{2}}+4y+4 $

Combining similar terms and simplifying, we get:
⇒ $ 5{{x}^{2}}-2xy+5{{y}^{2}}-76x-88y+506=0 $

The correct answer is B. $ 5{{x}^{2}}-2xy+5{{y}^{2}}-76x-88y+506=0 $ .

Note:The standard form, $ \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1 $ , of an ellipse in Cartesian coordinates assumes that the origin $ (0,\ 0) $ is the center of the ellipse, the x-axis is the major axis, the foci are the points $ F(\pm c,\ 0) $ and the vertices are $ V(\pm a,\ 0) $ .

The eccentricity of the ellipse is $ e=\dfrac{c}{a}=\sqrt{1-{{\left( \dfrac{b}{a} \right)}^{2}}} $ , for $ a>b $ .

The shifted form of the equation is $ \dfrac{{{\left( x-{{x}_{0}} \right)}^{2}}}{{{a}^{2}}}+\dfrac{{{\left(
y-{{y}_{0}} \right)}^{2}}}{{{b}^{2}}}=1 $ , where $ \left( {{x}_{0}},{{y}_{0}} \right) $ is the new shifted center.

In analytic geometry, the ellipse is defined as a quadric: the set of points $ (x,\ y) $ of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation $
a{{x}^{2}}+bxy+c{{y}^{2}}+dx+ey+f=0 $ , provided $ {{b}^{2}}-4ac<0 $ .