A hyperbola H:$$x29-y24=1$$ ntersects the circle, C:$$x2+y2-8x=0$$ at the points A and B.$$Statement-1$$: $$2x-5y+4=0$$ is a common tangentto both C and H. $$Statement-2$$: Circle on AB as a diameter passes through the centre of the hyperbola H.

Question

A hyperbola H:$$x29-y24=1$$ ntersects the circle, C:$$x2+y2-8x=0$$ at the points A and B.$$Statement-1$$: $$2x-5y+4=0$$ is a common tangentto both C and H. $$Statement-2$$: Circle on AB as a diameter passes through the centre of the hyperbola H.

Infinity Learn · Accepted Answer

Equation of a tangent with slope $$25$$ $$x29-y24=1$$ is $$y=25x+9$$×$$45-4$$⇒$$2x-5y+4=0Next$$ $$2x-5y+4=0$$ touches the circle (x-4)2+y2$$ $$=$$ $$16$$ if the length of the perpendicular from $$(4$$, $$0) on the line is $$4$$ which is true. Hence $$statement-1$$ is true. In $$statement-2$$, let A be (3sec$$θ,$$2tan$$θ$$)A$$ lies on the circle $$x2+y2-8x=0$$⇒$$13sec2$$θ$$-24sec$$θ$$-4=0$$⇒$$sec$$θ$$=2$$⇒$$tan$$θ$$=$$±$$3So$$ the coordinate of A are $$(6$$,$$23) and of B $$are(6$$,$$-23)$$ and equation of the circle on AB as diameter is (x-6)(x-6)+(y-23)(y+23)=0$$⇒$$x2+y2-12x+24=0$$ which does not pass through the centre $$(0$$, $$0) of the hyperbola. Thus $$statement-2$$ is false.

A hyperbola $H : \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ ntersects the circle, $C : x^{2} + y^{2} - 8 x = 0$ at the points A and B.
Statement-1: $2 x - \sqrt{5} y + 4 = 0$ is a common tangentto both C and H.
Statement-2: Circle on AB as a diameter passes through the centre of the hyperbola H.

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A hyperbola H:x29-y24=1 ntersects the circle, C:x2+y2-8x=0 at the points A and B.Statement-1: 2x-5y+4=0 is a common tangentto both C and H. Statement-2: Circle on AB as a diameter passes through the centre of the hyperbola H.

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A hyperbola $H : \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ ntersects the circle, $C : x^{2} + y^{2} - 8 x = 0$ at the points A and B.
Statement-1: $2 x - \sqrt{5} y + 4 = 0$ is a common tangentto both C and H.
Statement-2: Circle on AB as a diameter passes through the centre of the hyperbola H.