Parabola Question - None

Question

257. In a parabola $y^2 = 4ax$, two points $P$ and $Q$ are taken such that the tangents drawn to the parabola at these points meet at directrix in $R$. Focus of locus of circumcentre of $\triangle PQR$ will be:

Accepted Answer

Step 1: Key property – Tangents to parabola y² = 4ax drawn from any point on the directrix meet at right angles. So if R is on directrix and tangents from R touch at P and Q, then ∠PRQ = 90°.

Step 2: Since ∠PRQ = 90°, the circumcenter of △PQR is the midpoint of hypotenuse PQ, and it lies ON the chord of contact PQ (which is perpendicular to the axis through the focus).

Step 3: As P and Q move on the parabola (with their tangents meeting at different points on directrix), the chord of contact PQ varies. The circumcenter traces the locus of midpoints of all chords of contact.

Step 4: Using parametric analysis: if tangents at parameters t₁ and t₂ meet at directrix, the midpoint M of PQ traces a parabola y² = 4a(x - a). This is a parabola with the same focus (a, 0) as the original, but shifted left by distance a.

Step 5: The focus of the locus parabola y² = 4a(x - a) is at (2a, 0), which coincides with the focus of the original parabola y² = 4ax.

∴ Answer: C (Focus of original parabola)

Question:

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