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.
संबंधित प्रश्न
Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.
Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
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 points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
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 the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes is
(a) \[\left( 4, \frac{8}{3} \right)\]
(b) \[\left( - 4, \frac{8}{3} \right)\]
(c) \[\left( 4, - \frac{8}{3} \right)\]
(d) none of these
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
Find the equation of the normal lines to the curve 3x2 – y2 = 8 which are parallel to the line x + 3y = 4.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
Find the points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of the point
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
