Tangents drawn from - beta to the hyperbola x2-a2-y2-b2-1 make angle -1 and -2 with the x -axis-If tan-1tan-2-1- then -2- beta -2-

The correct option is C Equation of tangent is y = mx + √(a^2 m^2 b^2) If this tangent passes through (α, β) we get β = m α + √(a^2 m^2 b^2)⇒ (β m α)^2 ...

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Standard XII

Mathematics

Definition of Hyperbola

Tangents draw...

Question

Tangents drawn from $(α, b e t a)$ to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ make angle $θ_{1} a n d θ_{2}$ with the x-axis.If $t a n θ_{1} t a n θ_{2} = 1$ , then $(α^{2} - b e t a^{2})$ =

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Solution

The correct option is C
Equation of tangent is $y = m x + \sqrt{a^{2} m^{2} - b^{2}}$
If this tangent passes through $(α, β)$ we get $β = m α + \sqrt{a^{2} m^{2} - b^{2}} \Rightarrow (β - m α)^{2} = a^{2} m^{2} - b^{2}$
$\Rightarrow (α^{2} - a^{2}) m^{2} - 2 α β m + (β^{2} - b^{2}) = 0$
Now $t a n θ_{1}, θ_{2}$ are the roots of this equation
$∴ t a n θ_{1} . t a n θ_{2} = 1 \Rightarrow \frac{β^{2} - b^{2}}{α^{2} - a^{2}} = 1 \Rightarrow α^{2} - a^{2} = β^{2} + b^{2} \Rightarrow α^{2} - β^{2} = a^{2} + b^{2}$

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Tangents drawn from (α,beta) to the hyperbola x2a2−y2b2=1 make angle θ1 and θ2 with the x-axis.If tan θ1 tan θ2=1, then (α2−beta2) =

Tangents drawn from $(α, b e t a)$ to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ make angle $θ_{1} a n d θ_{2}$ with the x-axis.If $t a n θ_{1} t a n θ_{2} = 1$ , then $(α^{2} - b e t a^{2})$ =