Consider set $$S=(a$$,$$b)$$:(a+3)t2=3$$−a and $$bt2$$−$$4t+b=0$$, where t is a real parameter. Let C be a curve which is formed by all elements of set S where $$(a$$,$$b) is a point in $$R2$$. Tangents are drawn from the point $$P(3$$,$$4)$$ to the curve C touching the curve C at point Q and R. If the circumcentre of triangle PQR is $$($$α,β$$)$$, then the value of α$$+3$$β is

Question

Consider set $$S=(a$$,$$b)$$:(a+3)t2=3$$−a and $$bt2$$−$$4t+b=0$$, where t is a real  parameter. Let C be a curve which is formed by all elements of set S where $$(a$$,$$b) is  a point in $$R2$$. Tangents are drawn from the point $$P(3$$,$$4)$$ to the curve C touching  the curve C at point Q and R. If the circumcentre of triangle PQR is $$($$α,β$$)$$, then  the value of α$$+3$$β is

Infinity Learn · Accepted Answer

$$t2=3$$−$$a3+a=4t$$−bb⇒$$t=3b2(3+a)$$ put in eq. (1) ⇒$$4a2+9b2=36$$⇒$$x29+y24=1Equation$$ of tangent from $$Py=mx$$±$$9m2+4(4$$−$$3m)=$$±$$9m2+4$$⇒$$m=$$∞,$$m=12x=3$$,x−$$2y+5=0$$ point of contact $$Q(3$$,$$0)R$$−$$98$$,$$85$$ perpendicular bisector of PQ:$$y=2$$ perpendicular bisector of RQ:y−$$45=3x$$−$$35$$. circumcentre (1$$, $$2)⇒α$$+3$$β$$=7$$

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