If m1 and m2 are the slopes of the tangents to the hyperbola x2-25-y2-16- 1 which pass through the point 6- 2 -Fnd the values of m1-m2 and m1m2-

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Latus Rectum of Hyperbola

If m1 and ...

Question

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

m_{1} + m_{2} = \frac{24}{11}, m_{1} m_{2} = \frac{20}{11}

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m_{1} + m_{2} = \frac{4}{11}, m_{1} m_{2} = \frac{20}{11}

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m_{1} + m_{2} = \frac{20}{11}, m_{1} m_{2} = \frac{20}{11}

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m_{1} + m_{2} = \frac{24}{11}, m_{1} m_{2} = \frac{19}{11}

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Solution

The correct option is A $m_{1} + m_{2} = \frac{24}{11}, m_{1} m_{2} = \frac{20}{11}$
Equation of tangent to the given hyperbola is given by,
$y = m x + \sqrt{a^{2} m^{2} - b^{2}}$
Given it passes through $(6, 2)$
$\Rightarrow 2 = 6 m + \sqrt{25 m^{2} - 16}$
$\Rightarrow (2 - 6 m)^{2} = 25 m^{2} - 16 \Rightarrow 11 m^{2} - 24 m + 20 = 0$
$∴ m_{1} + m_{2} = \frac{24}{11}, m_{1} m_{2} = \frac{20}{11}$

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If m1 and m2 are the slopes of the tangents to the hyperbola x225−y216=1 which pass through the point (6,2) .Fnd the values of m1+m2 and m1m2.

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