A circle C whose radius is $$1$$ unit touches the $$x-axis$$ at point A. The center Q of C lies in the first quadrant. The tangent from the origin O to the circle touches it at T and a point P lies on it such that ∆OAP is a $$right-angled$$ triangle at A and its perimeter is $$8$$ units.The length of PQ isThe equation of circle C isThe equation of tangent OT is

Question

A circle C whose radius is $$1$$ unit touches the $$x-axis$$ at point A. The center Q of C lies in the first quadrant. The tangent from the origin O to the circle touches it at T and a point P lies on it such that ∆OAP is a $$right-angled$$ triangle at A and its perimeter is $$8$$ units.The length of PQ isThe equation of circle C isThe equation of tangent OT is

Infinity Learn · Accepted Answer

Given $$QT=QA=1$$.Let PQ $$=$$ x.Then $$PT=x2$$−$$1$$ΔTQP and △APO are similar triangles. Then, $$OT=OA=x+1x2$$−$$1or$$   $$1+x+2(x+1)x2$$−$$1+x2$$−$$1=8or$$   $$x=53AP=83$$,$$OP=163$$ Let   ∠$$AOP=2$$θ.Then, $$sin$$⁡$$2$$θ$$=12$$ From ΔOAQ, $$tan$$⁡θ$$=1OA$$ or   $$OA=1tan$$ ⁡θ  Also,      $$sin$$ ⁡$$2$$θ$$=2tan$$,⁡θ$$1+tan2$$ ⁡θ$$=12$$ ∴ $$tan$$⁡ θ$$=2$$−$$3Hence$$, $$OA=2+3Hence$$, the equation of circle is $${x$$−$$(2+3)}2+(y$$−$$1)2=1$$.The equation of tangent OT is x−$$0cos$$ ⁡$$2$$θ$$=y$$−$$0sin$$ ⁡$$2$$θ or   x−$$3y=0$$

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