Let RS be the diameter of the circle $$x2+y2=1$$, where S is the point (1$$,$$0) . Let P be a variable point (other$$ than R and $$S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

Question

Let RS be the diameter of the circle $$x2+y2=1$$, where S is the point (1$$,$$0) . Let P be a variable point (other$$ than R and $$S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

Infinity Learn · Accepted Answer

Let $$P($$$$cos$$α,$$sin$$α$$)$$;$$s(1$$,$$0)$$ and  RS is X $$-axis$$ and Tangent at P is xcosα$$+ysin$$α$$=1$$→$$1tangent$$ at $$S1$$,$$0$$ is $$x=1$$ →$$2solve$$ $$1$$,$$2$$∴$$Q1$$,$$1-$$$$cos$$α$$sin$$α , now  Normal at P is xsinα$$-ycos$$α$$=0$$ →$$3$$ Line through Q parallel to x $$-axis$$ is $$y=1-$$$$cos$$α$$sin$$α  , solve with $$3$$,∴Ecosα$$1+$$$$cos$$α,$$1-$$$$cos$$α$$sin$$α$$=(h$$,$$k)$$ we have to find  Locus of E,   now  $$k=1-$$$$cos$$α$$sin$$α$$=$$$$tan$$α$$2h=1$$−$$11+$$$$cos$$α$$=1$$−$$12sec2$$α$$2=1$$−$$121+k2$$∴locus of E  is $$x=12$$−$$12y2$$ ∴$$y2=1$$−$$2x$$

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