Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
उत्तर
`x=3costheta-cos^3theta,` `y=3sintheta-sin^3theta`
`rArr(dx)/(d theta)=-3sintheta +3cos^2thetasintheta `
And
`rArr(dy)/(d theta)=3costheta-3sin^2thetacostheta`
`rArr(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(3costheta-3sin^2thetacos theta)/(-3sintheta+3cos^2thetasintheta)=(costheta(1-sin^2theta))/(-sintheta(1-cos^2theta))=cos^3theta/-sin^3theta=-tan^3theta`
So equation of the tangent at θ is
`y-3sintheta+sin^3theta=-tan^3theta(x-3costheta+cos^3theta)`
`rArr4(ycos^3theta-xsin^3theta)=3sin4theta`
So equation of normal at θ is
`y-3sintheta+sin^3theta=1/tan^3theta(x-3costheta+cos^3theta)`
`rArrycos^3theta-xcos^3theta=3sin^4theta-sin^6theta-3cos^4theta+cos^6theta`
`rArr ysin^3theta-xcos^3theta=3sin^4theta-sin^6theta-3cos^4theta+cos^6theta`
APPEARS IN
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 the equation of all lines having slope −1 that are tangents to the curve `y = 1/(x -1), x != 1`
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t = `pi/4`
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
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 equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
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 __________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
