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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+3β is

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answer is 7.

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Detailed Solution

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+3β=7

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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+3β is