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  • If  and <img alt="\mathrm{m_{2}}" src="https://entrancecorner.oncodecogs.co

If \mathrm{m_{1}} and \mathrm{m_{2}} are the slopes of the tangents to the hyperbola \mathrm{\frac{x^2}{25}-\frac{y^2}{16}=1} which pass through the point \mathrm{(6, 2)}, then

Option: 1

\mathrm{m_1+m_2=-\frac{24}{11}}


Option: 2

\mathrm{m_1 m_2=\frac{20}{11}}


Option: 3

\mathrm{m_1+m_2=\frac{48}{11}}


Option: 4

\mathrm{m_1 m_2=\frac{11}{20}}


Answers (1)

best_answer

The line through (6, 2) is \mathrm{y-2=m(x-6) \Rightarrow y=m x+2-6 m}
Now, from condition of tangency \mathrm{(2-6 m)^2=25 m^2-16}
\mathrm{\Rightarrow \quad 36 m^2+4-24 m-25 m^2+16=0 \Rightarrow 11 m^2-24 m+20=0}
Obviously, its roots are \mathrm{m_{1}} and \mathrm{m_{2}} , therefore \mathrm{m_1+m_2=\frac{24}{11} \text { and } m_1 m_2=\frac{20}{11}}

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