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.
Advertisements
उत्तर
Hence, the perpendicular distance of the normal from the origin is constant.
APPEARS IN
संबंधित प्रश्न
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
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).
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
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 slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
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 ?
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 values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find the points on the curve y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
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 2a2y = x3 − 3ax2 where the tangent is parallel to x-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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
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 tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
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 coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
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 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 slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
Find the equation of the normal lines to the curve 3x2 – y2 = 8 which are parallel to the line x + 3y = 4.
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 two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
