If the line 2 x + 6 y = 2 touches the hyperbola x 2 - 2 y 2 = 4 , then the point of contact is [2004]

Question

If the line $2 x + \sqrt{6} y = 2$ touches the hyperbola $x^{2} - 2 y^{2} = 4,$ then the point of contact is [2004]

Smart Achievers · Accepted Answer

(4)

$Equation of tangent to hyperbola x^{2} - 2 y^{2} = 4 at any point (x_{1}, y_{1}) is$

$x x_{1} - 2 y y_{1} = 4$ ....(i)

$But the given tangent is 2 x + \sqrt{6} y = 2$

$On comparing equation (i) with 2 x + \sqrt{6} y = 2$

$i.e., 4 x + 2 \sqrt{6} y = 4, we get$

$x_{1} = 4 and - 2 y_{1} = 2 \sqrt{6}$ $\Rightarrow (4, - \sqrt{6}) is the required point of contact.$

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