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Q.

Let a line parallel to the axis of the parabola y2=4x  be drawn through point P(α−5,α),(α≠0) to meet the parabola at Q. Equation of tangent at Q is  x−2y+4=0. Let T(≠Q)  be a point on the tangent, and M, N be feet of perpendiculars on SQ and directrix respectively from point T. (S is focus), then which of the following option(s) is/are INCORRECT?

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a

α is equal to  −4

b

If QM=3, then TN is 2

c

If two tangents are drawn from point P to the parabola touching it at A and B and Circumcentre of ΔPAB is at (β,γ), then γ  is  −1

d

If orthocenter of ΔPAB (as defined in option C) is at (a,b), then a+b=3 

answer is A, C.

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Detailed Solution

Let Q≡(α24,α) lies on x-2y+4=0 ⇒α=4
Δs TMQ and TRQ are congruent ⇒QM=QR⇒SM=TN=SQ−QM=5−3=2  
Now, P=(-1,4) which lies on directrix  Two tangents are perpendicular 
⇒P is orthocentre and mid-point of AB as circumcentre which has same y coordinate as that of P
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