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Let P (h,K) be any point on curve y=f...

Let P (h,K) be any point on curve `y=f(x).` Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m `=(dy)/(dx)]_()(h,k))` = shope of tangent.

(i) Length of tangent =PT `=|K| sqrt(1+(1)/(m^(2)))`
(ii) Length of Normal =PN +`|K| sqrt(1+m^(2))`
(iii) Length subtangent = Projection of segment PT on x-axis `=TM =|(k)/(m)|`
(iv) Length of subnormal =Projection of line segment PN on x-axis =MN `=|Km|`
Find the product of length of tangent and length of normal for the curve `y=x^(3)+3x^(2)+4x-1` at point x=0

A

`(17)/(4)`

B

`(sqrt(15))/(4)`

C

`17`

D

`(4)/(sqrt(17 ))`

Text Solution

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The correct Answer is:
A
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Knowledge Check

  • Let P (h,K) be any point on curve y=f(x). Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m =(dy)/(dx)]_()(h,k)) = shope of tangent. (i) Length of tangent =PT =|K| sqrt(1+(1)/(m^(2))) (ii) Length of Normal =PN + |K| sqrt(1+m^(2)) (iii) Length subtangent = Projection of segment PT on x-axis =TM =|(k)/(m)| (iv) Length of subnormal =Projection of line segment PN on x-axis =MN =|Km| Find length of subnormal to x=sqrt(2) cos t,y =- 3sin t " at " t= (-pi)/(4).

    A
    `(2)/(9)`
    B
    `1`
    C
    `(7)/(2)`
    D
    `(9)/(2)`

  • Let P (h,K) be any point on curve y=f(x). Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m =(dy)/(dx)]_()(h,k)) = shope of tangent. (i) Length of tangent =PT =|K| sqrt(1+(1)/(m^(2))) (ii) Length of Normal =PN + |K| sqrt(1+m^(2)) (iii) Length subtangent = Projection of segment PT on x-axis =TM =|(k)/(m)| (iv) Length of subnormal =Projection of line segment PN on x-axis =MN =|Km| Determine 'p' such that the length of the subtangent nad subnormal is equal for the curve y=e^(px) +px at the point (0,1)

    A
    `+- 1`
    B
    `+- 2`
    C
    `+-(1)/(2)`
    D
    `+-(1)/(4)`

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