wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Tangent at a point P1, other than 0,0 on the curve y=x3 meets the curve again at P2. The tangent at P2 meets the curve at P3 and so on. Then the abscissa of P1, P2, P3… are in:


A

A.P.

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

G.P.

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C

H.P.

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is B

G.P.


Explanation for correct option

Finding if the abscissae are in GP

The given curve is y=x3

Then, dydx=3x2

Let the coordinates of the point P1be x1,y1

Then the slope of the tangent to the curve at

P1x1,y1=dydx(x1,y1)=3x12

So the equation of the tangent to the curve at P1is

y-y1=3x12x-x1x3-x13-3x12(x-x1)=0x-x1x2+xx1+x12-3x12(x-x1)=0x-x1x2+xx1+x12-3x12=0x-x1x2+xx1-2x12=0x-x1x2-xx1+2xx1-2x12=0x-x1xx-x1+2x1x-x1=0x-x1x-x1x+2x1=0x=x1orx=-2x1x=-2x1.

Now,

Let the coordinates of the point P2be x2,y2

So,

x2=-2x1,y2=x23=-2x13=-8x13

This shows that the tangent at the point P1x1,y1 meets the curve at P22x1,-8x13.

Again the slope of the tangent to the curve at

P2=dydxx2,y2=3x22

The equation of a tangent to the curve at P2 is

y-y2=3x22x-x2x-x2x-x2x+2x2=0x=-2x2.

since,

x3=-2x2=-2×-2x1=4x1andy3=x33=4x13=64x13.So,P3=x3,y3=4x1,64x13

Hence the abscissae are x1,-2x1,4x1,...

Therefore, the abscissae are in GP in the common ratio -2

Therefore, the correct answer is option B.


flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon