A) \[4{{x}^{2}}+{{y}^{2}}-4xy+8x+46y-71=0\]
B) \[3{{x}^{2}}+2{{y}^{2}}-7xy+3x++9y-15=0\]
C) \[2{{x}^{2}}+3{{y}^{2}}-5xy+9x+7y-14=0\]
D) none of these
Correct Answer: A
Solution :
Let \[({{x}_{1}},{{y}_{1}})\] be the co-ordinates of tRe point of intersection of the axis and the directix. Then the vertex is the mid-points of tlie line segment joining\[({{x}_{1}},{{y}_{1}})\] and the focus \[(1,-1)\] \[\therefore \] \[\frac{{{x}_{1}}+1}{2}=2\]and \[\frac{y+(-1)}{2}=1\] \[\Rightarrow \]\[{{x}_{1}}=3,{{y}_{1}}=3\] Thus, the directrix meets the axis at (3, 3), let \[{{m}_{1}}=\]slope of the line joining the focus and the vertex \[=\frac{1+1}{2-1}=2.\] Slope of directrix \[=-\frac{1}{2}.\]Thus the directrix passes through (3, 3) and has slope \[-1/2.\] So, its equation is \[\Rightarrow \]\[y-3=-\frac{1}{2}(x-3)\Rightarrow x+2y-9=0.\] Distance of P from the focus = distance of P from the director \[\Rightarrow \]\[\sqrt{{{(x-1)}^{2}}+{{(y+1)}^{2}}}=\left| \frac{x+2y-9}{\sqrt{x+4}} \right|\] \[\Rightarrow \]\[{{(x-1)}^{2}}+{{(y+1)}^{2}}=\frac{{{(x+2y-9)}^{2}}}{5}\] \[\Rightarrow \]\[4{{x}^{2}}+{{y}^{2}}-4xy+8x+46y-71=0\]
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