The length of the minor axis (along the y-axis) of an ellipse in the standard form is 34. If this ellipse touches the line x+6y=8, then its eccentricity is:
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Official solution
Let the equation of ellipse
\[{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\], (a>b)
Given 2b = \[{4 \over {\sqrt 3 }}\]
\[ \Rightarrow \] b = \[{2 \over {\sqrt 3 }}\]
We know, Equation of tangent y = mx \[ \pm \] \[\sqrt {{a^2}{m^2} + {b^2}} \] ....(1)
Given tangent is x + 6y = 8
\[ \Rightarrow \] y = \[ - {1 \over 6}x + {8 \over 6}\] .....(2)
By comparing (1) and (2),
m = \[ - {1 \over 6}\] and a2m2+b2 = \[{{16} \over 9}\]
\[ \Rightarrow \] \[{{a^2}\left( {{1 \over {36}}} \right) + {4 \over 3}}\] = \[{{16} \over 9}\]
\[ \Rightarrow \] a2=16
\[ \therefore \] e = \[\sqrt {1 - {{{b^2}} \over {{a^2}}}} \]
= \[\sqrt {1 - {{{4 \over 3}} \over {16}}} \]
= \[\sqrt {{{11} \over {12}}} \] = \[{1 \over 2}\sqrt {{{11} \over 3}} \]