Advertisements
Advertisements
प्रश्न
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Advertisements
उत्तर
Given:
x=3 cost−cos3t
y=3 sint−sin3t
Slope of the tangent, `dy/dx=(dy/dt)/(dx/dt)=(3cost-3sin^2tcost)/(-3sint+3cos^2tsint)`
`=(3cost[cos^2t])/(-3sint[sin^2t])`
`dy/dx=(-cos^3t)/sin^3t`
∴Slope of the normal
`=sin^3t/cos^3 t`
The equation of the normal is given by
`(y-(3sint-sin^3t))/(x-(3cost-cos^3t))=sin^3t/cos^3t`
`=>ycos^3t-3sint cos^3t +sin^3tcos^3t=xsin^3t-3costsin^3t+sin^3tcos^3t`
`=>ycos^3t-xsin^3t=3(sintcos^3t-costsin^3t)`
`=>ycos^3t-xsin^3t=3sintcost(cos^2t-sin^2t)`
`=>ycos^3t-xsin^3t=3/2sin2tcos2t=3/4sin4t`
`=>4(ycos^3t-xsin^3t)=3sin4t`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
(A) (1, 2)
(B) (2, 1)
(C) (1, −2)
(D) (−1, 2)
Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
