Question

If the tangent to the curve y = x³ at the point P(t, t³) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1:2 is ______.

पर्याय

• –2t³

• –t³

• 0

• 2t³

Answer 1

If the tangent to the curve y = x³ at the point P(t, t³) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1:2 is `underlinebb(-2"t"^3)`.

Explanation:

Given: Tangent to the curve y = x³ at the point

P(t, t³) meets the curve again at Q

Curve is y = x³  ...(i)

Differentiate both sides w.r.t.x

`("dy")/("d"x)` = 3x²

⇒ `|("dy")/("d"x)|_(("t""," "t"^3))` = 3t² = Slope of tangent at P

Equation of straight line having slope m and passing through (x₁, x₁) is y – y₁ = m(x – x₁)

Here, m = 3t², (x₁, y₁) = (t, t³)

∴ Equation of tangent at P(t, t³) is

(y – t³) = 3t²(x – t)  ...(ii)

Putting value of y from equation (i), we get

x³ –  t³ = 3t²(x –  t)

⇒ x² + xt + t² = 3t²

⇒ x² + xt – 2t² = 0

⇒ (x – t)(x + 2t) = 0

⇒ x₁ = t, x² = –2t,

From, y = x³

y₁ = t³, y₂ = –8t³

∴ Point Q is (–2t, –8t³).

Use section formula, to find R which divides PQ internally in the ratio 1:2

∴ R = `((1 xx (-2"t") + 2"t")/(1 + 2), (1 xx (-8"t"^3) + 2"t"^3)/(1 + 2))` ≡ (0, –2t³)

So, the coordinate of required point is –2t³.