A tangent at a point P 1 other than (0, 0) on the curve y = x 3 meets the curve again at P 2 . The tangent at P 2 meets the curve at P 3 and so on. The abscissae of P 1 P 2 ……… P n form a G.P. Find the ratio of 
.
Text Solution
Verified by Experts0016
Ans. 0016
Sol. Any point on curve y = x 3 is of form (t, t 3 ) and
= 3t 2 Equation of tangent at (t, t
3 ) is
y – t 3 = 3t 2 (x – t)
⇒ 3t 2 x – y – 2t 3 = 0 …(i)
The intersection of (i) with y = x 3 is given by
3t 3 x – x 3 – 2t 3 = 0
3x (t 2 – x 2 ) + 2(x 3 – t 3 ) = 0
(x – t) [–3x 2 – 3xt + 2x 2 + 2t 2 + 2xt] = 0
(x – t) (–x 2 – xt + 2t 2 ) = 0
x = t or x = 2t
Therefore, the tangent at x = t meets y = x 3 at point P 2
(t 2 , 
) when t 2 = – 2t
The abscissae of P 1 , P 2 , …. P n are t, – 2t … (– 2) n – 1 t(G.P.)
Area of Δ P 1 P 2 P 3 = 
| 
|
= 
t 4 | 
|= 
t 4 | 
|
(from R 2 → R 2 + 2R 1 , R 3 → R 3 – 4R 1 )
= 
t 4 × 162 = 81 t 4 area of Δ P 2 P 3 P 4 = 81 
= 
= 
= 
∴ 
= 16
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems