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Question

The tangent at any point of the curve x=at3,y=at4 divides the abscissa of the point of contact in the ratio

A
1:4
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B
3:2
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C
1:3
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D
3:1
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Solution

The correct option is A 1:4
Given:
x=at2
y=at4

Then,
t3=xa,t4=ya

t=(xa)1/3t=(ya)1/4

(xa)1/3=(ya)1/4

(xa)13×412=(ya)14×12

x4a4=y3a2

y3=x4a at P(h,k)

Also relation k3=h4a

Now by differentiation of after equation we get

3y2dydx=4x2a

dydx=4x33ay2

Now MT=dydx=4x33ay2

Now we will find slope of tangent

MT/P(h,x)=4h33ax2

equation of tangent P(h,k)

yk=4h33ak2(xh)

Let y=0

Then, k=4h33ak2(xh)

3ak34h3=xh

Earlier we found that k3=h4a

Then ax3=h4

3h44h2=xh

3h4+h=x

x=h4

xh=14

Hence option (a) 1:4 is the correct choice

1496738_1244227_ans_0a282e0fbce94a80bd120e501512f784.JPG

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