Question

Let a variable point A be lying on the directrix of parabola y² = 4ax (a > 0). Tangents AB and AC are drawn to the curve where B and C are points of contact of tangents. The locus of centroid of ΔABC is a conic whose length of latus rectum is λ, then `λ/"a"` is equal to ______.

Options

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Answer 1

Let a variable point A be lying on the directrix of parabola y² = 4ax (a > 0). Tangents AB and AC are drawn to the curve where B and C are points of contact of tangents. The locus of centroid of ΔABC is a conic whose length of latus rectum is λ, then `λ/"a"` is equal to 3.

Explanation:

Consider, `"A"(−"a", "a"("t"_1 + "t"_2)) , "B"("at"_1^2, 2"at"_1), "C"("at"_2^2, 2"at"_2)`

Let (h, k) is centroid of ΔABC

h = `("a"("t"_1^2 + "t"_2^2) - "a")/3`

and k = a(t₁ + t₂)

⇒ 3h = `"a"(("t"_1 + "t"_2)^2 + 2) - "a"` ...[∴ t₁t₂ = –1]

⇒ 3h = `"a"("k"^2/"a"^2 + 1)`

⇒ 3h = `"k"^2/"a" + "a"`

⇒ k² = `3"a"("h" - "a"/3)`

⇒ λ = 3a  ...[λ = Latus rectum]

⇒ `λ/"a"` = 3