मराठी

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
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 6: Application of Derivatives - Exercise 6.6 [पृष्ठ २४२]

APPEARS IN

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

व्हिडिओ 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.


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


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

y = x3 at (1, 1)


Find the equation of the normals to the curve y = x3 + 2+ 6 which are parallel to the line x + 14y + 4 = 0.


Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]


The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is

(A) 1

(B) 2

(C) 3

(D) 1/2


Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.


Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 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 point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?


Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is  parallel to y-axis ?


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 and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?


Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?


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  \[y = \sqrt{3x - 2}\] 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 angle of intersection of the following curve y = 4 − x2 and y = x2 ?


Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?


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


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 y2 = x where tangent makes 45° angle with x-axis is ____________________ .


The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .


The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .


The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .


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


Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.


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


The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.


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 line y = x + 1 is a tangent to the curve y2 = 4x at the point


If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then


The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is


The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are


An edge of variable cube is increasing at the rate of 3 cm/s. The volume of the cube increasing fast when the edge is 10 cm long is ______ cm3/s.


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 ______.


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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×