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?

Let Q≡(α24,α) lies on x-2y+4=0 ⇒α=4Δs 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

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?

Let Q≡(α24,α) lies on x-2y+4=0 ⇒α=4Δs 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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Let a line parallel to the axis of the parabola $y^{2} = 4 x$ be drawn through point $P (α - 5, α), (α \neq 0)$ to meet the parabola at Q. Equation of tangent at Q is $x - 2 y + 4 = 0$ . Let $T (\neq Q)$ 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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$α$ is equal to $- 4$

If QM=3, then TN is 2

If orthocenter of $Δ P A B$ (as defined in option C) is at $(a, b),$ then $a + b = 3$

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

answer is A, C.

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Detailed Solution

Let $Q \equiv (\frac{α^{2}}{4}, α)$ lies on x-2y+4=0 $\Rightarrow α = 4$
$Δ s$ TMQ and TRQ are congruent $\Rightarrow Q M = Q R$ $\Rightarrow S M = T N = S Q - Q M = 5 - 3 = 2$
Now, P=(-1,4) which lies on directrix Two tangents are perpendicular
$\Rightarrow$ P is orthocentre and mid-point of AB as circumcentre which has same y coordinate as that of P
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