Hyperbola in conic sections

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Question

$The absolute value of slope of common tangents to$ $parabola y^{2} = 8 x and hyperbola 3 x^{2} - y^{2} = 3 is$

Moderate

Solution

$Tangent to y^{2} = 8 x \Rightarrow y = m x + \frac{2}{m}$ $\to (1)$ $(\sin c e y = m x + \frac{a}{m})$
$g i v e n h y p e r b o l a e q u a t i o n i s \frac{x^{2}}{1} - \frac{y^{2}}{3} = 1$
$Tangent to \frac{x^{2}}{1} - \frac{y^{2}}{3} = 1 is y = m x \pm \sqrt{m^{2} - 3} → (2)$ $(\sin c e y = m x \pm \sqrt{a^{2} m^{2} - b^{2}})$
$(1), (2) r e p r e s e n t s s a m e l i n e$ .
$\Rightarrow \frac{2}{m} = \sqrt{m^{2} - 3} \Rightarrow 4 = m^{2} (m^{2} - 3)$ $\Rightarrow$ $m = \pm 2$

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Similar Questions

Statement-1: The locus of the point of intersection of the tangents that are at right angles to the hyperbola $\frac{x^{2}}{36} - \frac{y^{2}}{16} = 1 is the circle x^{2} + y^{2} = 52$

Statement-2: Perpendicular tangents to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ interest on the director circle $x^{2} + y^{2} = a^{2} - b^{2} (a^{2} > b^{2})$ of the hyperbola.