A) \[\frac{\sqrt{5}}{2}\]
B) \[\frac{\sqrt{3}}{2}\]
C) \[\frac{2}{\sqrt{5}}\]
D) \[\frac{2}{\sqrt{3}}\]
Correct Answer: A
Solution :
Equation of parabola, \[{{y}^{2}}=6x\] \[\Rightarrow \]\[{{y}^{2}}=4\times \frac{3}{2}x\] \[\therefore \]Focus \[=\left( \frac{3}{2},0 \right)\] Let equation of chord passing through focus be \[ax+by+c=0\] ...(1) Since chord is passing through \[\left( \frac{3}{2},0 \right)\] \[\therefore \]Put \[x=\frac{3}{2},y=0\] in eqn (1), we get \[\frac{3}{2}a+c=0\] \[\Rightarrow \] \[c=-\frac{3}{2}a\] ?(2) distance of chord from origin is \[\frac{\sqrt{5}}{2}\] \[\frac{\sqrt{5}}{2}=\left| \frac{a(0)+b(0)+c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right|=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] Squaring both sides \[\frac{5}{4}=\frac{{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] \[\Rightarrow \] \[{{a}^{2}}+{{b}^{2}}=\frac{4}{5}{{c}^{2}}\] Putting value of c from (2), we get \[{{a}^{2}}+{{b}^{2}}=\frac{4}{5}\times \frac{9}{4}{{a}^{2}}\] \[{{b}^{2}}=\frac{4}{5}{{a}^{2}}-{{a}^{2}}=\frac{4}{5}{{a}^{2}}\] \[\frac{{{a}^{2}}}{{{b}^{2}}}=\frac{5}{4},\frac{a}{b}=\pm \frac{\sqrt{5}}{2}\] Slope of chord, \[\frac{dy}{dx}=-\frac{a}{b}=-\left( \frac{\pm \sqrt{5}}{2} \right)=\overline{+}\frac{\sqrt{5}}{2}\]
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