If the chords of contact of the tangents from a point on the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$a2$$ to the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$b2$$ touch the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$c2$$, then the roots of the equation $$ax2$$ $$+$$ $$2bx$$ $$+$$ c $$=$$ $$0$$, are

Question

If the chords of contact of the tangents from a point on the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$a2$$ to the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$b2$$ touch the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$c2$$, then the roots of the equation $$ax2$$ $$+$$ $$2bx$$ $$+$$ c $$=$$ $$0$$, are

Infinity Learn · Accepted Answer

Let $$P(x1$$, $$y1)$$ be a point on $$x2+y2=a2$$. Then,          $$x12+y12=a2$$                                      …(i)Let$$ QR be the chord of contact of tangents drawn from P $$(x1$$ , $$y1) to the circle $$x2$$ $$+$$ $$y2$$ $$=$$ $$b2$$. Then, the equation QR is      $$xx1+yy1=b2$$                                    …(ii)This$$ touches the circle $$x2+y2=c2$$∴ $$0x1+0y1$$−$$b2x12+y12=c$$⇒$$b2=ac$$          [Using: $$(i)]Let D be the discriminant of $$ax2$$ $$+$$ $$2bx$$ $$+$$ c $$=$$ $$0$$. Then,       $$D=4b2$$−$$ac=0$$           ∵$$b2=acHence$$, the roots of the given equal are real and equal.

If the chords of contact of the tangents from a point on the circle $x^{2} + y^{2} = a^{2}$ to the circle $x^{2} + y^{2} = b^{2}$ touch the circle $x^{2} + y^{2} = c^{2},$ then the roots of the equation $a x^{2} + 2 b x + c = 0,$ are

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If the chords of contact of the tangents from a point on the circle x2 + y2 = a2 to the circle x2 + y2 = b2 touch the circle x2 + y2 = c2, then the roots of the equation ax2 + 2bx + c = 0, are

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If the chords of contact of the tangents from a point on the circle $x^{2} + y^{2} = a^{2}$ to the circle $x^{2} + y^{2} = b^{2}$ touch the circle $x^{2} + y^{2} = c^{2},$ then the roots of the equation $a x^{2} + 2 b x + c = 0,$ are