Explanation
\because \text{ Length of latus rectum = 4a}
\text{ Hence Length of latus rectum of given parabola is} = \dfrac{1}{a}
\textbf{Hence, Option 'C' is correct}
Equation of circle having centre of origin (0,0) and radius =r,
S_{1};x^{2}+y^{2}=r^{2}----(1)
\therefore f(m_{1},\dfrac{1}{m_{2}}),f(m_{2},\dfrac{1}{m_{2}}),--f(m_{4},\dfrac{1}{m_{4}})These points lie on S_{1}.
Let \displaystyle f(m, \frac{1}{m}) is point lie on S_{1},
m^{2}+\dfrac{1}{m^{2}}=r^{2}
m^{4}+1-r^{2}m^{2}=0
m^{4}-1-r^{2}m^{2}+1=0
m_{1},m_{2},m_{3} and m_{4} are roots of this equation
So, m_{1}m_{2}m_{3}m_{4} =1
136 (x^{2}+y^{2})=(5x+3y+7)^{2}
\Rightarrow \displaystyle (x-0)^2+(y-0)^2= \dfrac{1}{136}(5x+3y+7)^{2}
\Rightarrow \displaystyle (x-0)^2+(y-0)^2= \left( \frac{1}{2}\right)^2 \left(\frac{5x+3y+7}{\sqrt{34}} \right)^2
From the above form, we get focus of the conic has coordinates (0,0), and directrix has equation 5x+3y+7=0 and eccentricity e=\dfrac{1}{2}
Distance between focus and directrix =\dfrac{7}{\sqrt{34}}
Length of latus rectum=2\dfrac{b^2}{a}=2 \times e \times distance bet. focus and directrix=2 \times \dfrac{1}{2} \times \dfrac{7}{\sqrt{34}}
= \dfrac{7}{\sqrt{34}}
Hence, option B.
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