हिंदी

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. - Mathematics

Advertisements
Advertisements

प्रश्न

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.

उत्तर

Hence, the perpendicular distance of the normal from the origin is constant.

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 6: Application of Derivatives - Exercise 6.6 [पृष्ठ २४२]

APPEARS IN

एनसीईआरटी Mathematics [English] Class 12
अध्याय 6 Application of Derivatives
Exercise 6.6 | Q 5 | पृष्ठ २४२

वीडियो ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्न

Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.


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 = x4 − 6x3 + 13x2 − 10x + 5 at (1, 3)


Find the equations of the tangent and normal to the given curves at the indicated points:

x = cos ty = 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.


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 y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?


Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?


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 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 the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?


Find the equation of the tangent and the normal to the following curve at the indicated points  x = asect, y = btant at t ?


Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 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) ?


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 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?


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 \[\alpha\] \[-\] b2sin\[\alpha\] = p?


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 ?


Write the equation of the normal to the curve y = cos x at (0, 1) ?


Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?


The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .


The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .


The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.


Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.


Find the angle of intersection of the curves y = 4 – x2 and y = x2.


At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?


If the straight line x cosα + y sinα = p touches the curve `x^2/"a"^2 + y^2/"b"^2` = 1, then prove that a2 cos2α + b2 sin2α = p2.


The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.


The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.


The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.


The tangent to the curve y = 2x2 - x + 1 is parallel to the line y = 3x + 9 at the point ____________.


Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis. 


Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to


The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is


If (a, b), (c, d) are points on the curve 9y2 = x3 where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.


Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×