"The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is :"

Question

Accepted Answer

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$2{x^2} + 3{y^2} = 5$

Equation of tangent having slope m.

$y = mx\, \pm \,\sqrt {{5 \over 2}{m^2} + {5 \over 3}}$

which passes through $(1,3)$

$3 = m\, \pm \sqrt {{5 \over 2}{m^2} + {5 \over 3}}$

${5 \over 2}{m^2} + {5 \over 3} = 9 + {m^2} - 6m$

${3 \over 2}{m^2} + 6m - {{22} \over 3} = 0$

$9{m^2} + 36m - 44 = 0$

${m_1} + {m_2} = - 4,\,{m_1}{m_2} = - {{44} \over 9}$

${({m_1} - {m_2})^2} = 16 + 4 \times {{44} \over 9} = {{320} \over 9}$

Acute angle between the tangents is given by

$$\alpha = {\tan ^{ - 1}}\left| {{{{m_1} - {m_2}} \over {1 + {m_1}{m_2}}}} \right|$$

$$ = {\tan ^{ - 1}}\left| {{{{{8\sqrt 5 } \over 3}} \over {1 - {{44} \over 9}}}} \right|$$

$= {\tan ^{ - 1}}\left( {{{24\sqrt 5 } \over {35}}} \right)$

$\alpha = {\tan ^{ - 1}}\left( {{{24} \over {7\sqrt 5 }}} \right)$

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The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is :