question_answer

A point ratio of whose distance from a fixed point and line \[x=9/2\] is always 2 : 3. Then locus of the point will be [DCE 2005]

A)            Hyperbola                                 

B)            Ellipse

C)            Parabola                                    

D)            Circle

Correct Answer: B

Solution :

           In question, \[PS=\frac{2}{3}PM\] (Given)                    Focus \[S(-2,\,0)\],                    Equation of directrix \[2x-9=0\]                    \[{{(PS)}^{2}}=\frac{4}{9}{{(PM)}^{2}}\] Þ \[{{(h+2)}^{2}}+{{(k)}^{2}}=\frac{4}{9}{{\left( \frac{2h-9}{2} \right)}^{2}}\]                    Þ \[9[{{(h+2)}^{2}}+{{(k)}^{2}}]=\frac{4{{(2h-9)}^{2}}}{4}\]                    Þ \[9{{h}^{2}}+9{{k}^{2}}+36h+36=4{{h}^{2}}+81+36h\]                    Þ \[\frac{5{{h}^{2}}}{45}+\frac{9{{k}^{2}}}{45}=1\] Þ \[\frac{{{h}^{2}}}{9}+\frac{{{k}^{2}}}{5}=1\] Þ 1                    Locus of point P(h, k) is \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\], which is an ellipse.