The lengths of tangent, subtangent, normal and subnormal for the curve ` y=x^(2)+x-1 ` at (1,1) are A,B,C and D respectively, then their increasing order is

Options

A. B,D,A,C
B. B,A,C,D
C. A,B,C,D
D. B,A,D,C

Answer

D

Explanation

The equation of the curve is `y=x^(2)+x-1.`
` therefore (dy)/(dx) =2x+1 rArr ((dy)/(dx))_((1","1)) =3 `
Now ,
A=Length of the tangent at (1,1)
` rArr A=(sqrt(1+((dy)/(dx))^(2)))/((dy)/(dx))=(sqrt(1+3^(2)))/(3) =(sqrt(10))/(3) `
B= Length of the subtangent at (1,1)
` B=(y)/((dx)/(dy))=(1)/(3) `
C=Length of the normal at (1,1)
` rArr C=sqrt(1+((dy)/(dx))^(2))=sqrt(1+3^(2))=sqrt(10) `
D=Length of the subnormal at (1,1)
` rArr D=y(dy)/(dx)=1 xx 3 =3 `
Thus, we have `B lt A lt D lt C .`