If the tangent at a point P1 (other than (0, 0)) on the curve ax3−y+b=0 meets the curve again atP2 .The tangent at P2 meets the curve at P3 and so on. If the abscissa of P1,P2,P3,......Pn form a G.P. then (a, b) may be:

Book Online Demo

Check Your IQ

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

If the tangent at a point $P_{1}$ (other than (0, 0)) on the curve $a x^{3} - y + b = 0$ meets the curve again at $P_{2}$ .The tangent at $P_{2}$ meets the curve at $P_{3}$ and so on. If the abscissa of $P_{1}, P_{2}, P_{3}, ...... P_{n}$ form a G.P. then (a, b) may be:

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$(2, 7)$

$(3, 5)$

$(1, 0)$

$(4, 9)$

answer is A, B, C, D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Given Curve is $a x^{3} - y + b = 0$ $\Rightarrow y = a x^{3} + b$
Let point $P_{1}$ be $(t_{1}, a t_{1}^{3} + b)$
Slope of tangent = ${\frac{d y}{d x}|}_{P_{1}} = 3 a t_{1}^{2}$
$∴$ Equation of tangent is $y - (a t_{1}^{3} + b) = 3 a t_{1}^{2} (x - t_{1})$
$∴$ Tangent meets curve at $P_{2} (t_{2}, a t_{2}^{3} + b)$
$∴$ $(a t_{2}^{3} + b) - (a t_{1}^{3} + b) = 3 a t_{1}^{2} (t_{2} - t_{1})$
$\Rightarrow a (t_{2}^{3} - t_{1}^{3}) = 3 a t_{1}^{2} (t_{2} - t_{1})$
$\begin{array}{l} \Rightarrow t_{2}^{2} + t_{1}^{2} + t_{2} t_{1} = 3 t_{1}^{2} (∴ t_{1} \neq t_{2}) \\ \Rightarrow t_{2}^{2} + t_{2} t_{1} - 2 t_{1}^{2} = 0 \\ \Rightarrow (t_{2} + 2 t_{1}) (t_{2} - t_{1}) = 0 \end{array}$
$\Rightarrow t_{2} = - 2 t_{1}$
Similarly, $t_{3} = - 2 t_{2}$
$∴$ abscissae are in G.P. for all values of a and b.