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The point(s) on the curve y^3+\ 3x^2=1...

The point(s) on the curve `y^3+\ 3x^2=12 y` where the tangent is vertical, is(are) ? (a)`(+-4/(sqrt(3)),\ -2)` (b) `(+-\ sqrt((11)/3,\ )\ 1)` (c)`(0,\ 0)` (d) `(+-4/(sqrt(3)),\ 2)`

A

`(pm(4)/(sqrt(3)),-2)`

B

`(pmsqrt((pi)/(3)),1)`

C

(0, 0)

D

`(pm(4)/(sqrt(3)),2)`

Text Solution

Verified by Experts

The correct Answer is:
D
We have,
`y^(3)+3x^(2)=12y " …(i)" `
`rArr 3y^(2) (dy)/(dx)+6x=12(dy)/(dx) " " `[Diff. w.r.t. x]
`rArr 3(y^(2)-4) (dy)/(dx)= -6x rArr (dy)/(dx) = -(2x)/(y^(2)-4)`
At point(s) where the tangent(s) is (are) vertical, `(dy)/(dx)` is not defined .
`therefore y^(2) -4=0 rArr y= pm 2`.
From (i), we find that
`y=2 rArr x=pm(4)/(sqrt(3))`
and, `y = -2 rArr x^(2) =(-16)/(3)`, which is not possible.
Hence, the required points are `(pm 4//sqrt(3), 2)`
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OBJECTIVE RD SHARMA ENGLISH-TANGENTS AND NORMALS-Chapter Test
  1. The point(s) on the curve y^3+\ 3x^2=12 y where the tangent is ver...

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  2. The abscissa of the point on the curve ay^(2)=x^(3), the normal at whi...

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  3. If the curves (x^2)/(a^2)+(y^2)/(b^2)=1 and (x^2)/(l^2)-(y^2)/(m^2)=1c...

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  4. The length of normal at any point to the curve, y=c cosh(x/c) is

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  5. If the sub-normal at any point on y=a^(1-n)x^(n) is of constant length...

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  6. The angle of intersection of the curves y=x^(2), 6y=7-x^(3) at (1, 1),...

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  7. The slope of the tangent to the curve x=t^2+3t-8,\ \ y=2t^2-2t-5 at ...

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  8. What is the angle between these two curves x^3-3xy^2+2=0 and 3x^2y-y^3...

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  9. about to only mathematics

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  10. If y=4x-5 is a tangent to the curve y^(2)=px^(3)+q at (2, 3), then:

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  11. The curve y-e^(xy)+x=0 has a vertical tangent at the point:

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  12. The tangent to the curve given by x = e^(t) cos t y = e^(t) " sin t ...

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  13. The length of the normal at t on the curve x=a(t+sint), y=a(1-cos t), ...

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  14. For the parabola y^(2)=4ax, the ratio of the subtangent to the absciss...

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  15. The length of the subtangent to the curve sqrt(x) +sqrt(y)=3 at the po...

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  16. Find the euation of normal to the curve x=a( cos theta + theta sin th...

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  17. Tangents ar drawn to y= cos x from origin then points of contact for t...

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  18. If m denotes the slope of the normal to the curve y= -3 log(9+x^(2)) a...

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  19. If m be the slope of the tangent to the curve e^(2y) = 1+4x^(2), then

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  20. If the curve y=ax^(3) +bx^(2) +c x is inclined at 45^(@) to x-axis at...

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  21. If the curve y=ax^(2)+bx+c passes through the point (1, 2) and the lin...

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