$$Statement-1$$: The locus of the point of intersection of the tangents that are at right angles to the hyperbola $$x236-y216=1$$ is the circle $$x2+y2=52Statement-2$$: Perpendicular tangents to the hyperbola $$x2a2-y2b2=1$$ interest on the director circle $$x2+y2=a2-b2a2$$$$b2$$ of the hyperbola.

Question

$$Statement-1$$: The locus of the point of intersection of the tangents that are at right angles to the hyperbola $$x236-y216=1$$ is the circle $$x2+y2=52Statement-2$$: Perpendicular tangents to the hyperbola $$x2a2-y2b2=1$$ interest on the director circle  $$x2+y2=a2-b2a2$$>$$b2$$ of the hyperbola.

Infinity Learn · Accepted Answer

Equation of a tangent to the hyperbola is y $$=$$ $$mx+a2m2-b2$$ and the equation of the tangent perpendicular to it is $$y=-1mx+a2m2-b2$$.Eliminating m we get the required locus as $$(y-mx)2$$ $$+(x+my)2=a2m2-b2+a2-b2m2$$⇒$$x2+y2=a2-b2$$ and the $$statement-2$$ is true using which $$statement-1$$ is false.

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Statement-1: The locus of the point of intersection of the tangents that are at right angles to the hyperbola x236-y216=1 is the circle x2+y2=52Statement-2: Perpendicular tangents to the hyperbola x2a2-y2b2=1 interest on the director circle x2+y2=a2-b2a2>b2 of the hyperbola.

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