English

Find the Equations of the Tangent and Normal to the Hyperbola `X^2/A^2 - Y^2/B^2` At the Point `(X_0, Y_0)` - Mathematics

Advertisements
Advertisements

Question

Find the equations of the tangent and normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`

Solution

shaalaa.com
  Is there an error in this question or solution?
Chapter 6: Application of Derivatives - Exercise 6.3 [Page 213]

APPEARS IN

NCERT Mathematics [English] Class 12
Chapter 6 Application of Derivatives
Exercise 6.3 | Q 24 | Page 213

RELATED QUESTIONS

Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.


Find the slope of the tangent to curve y = x3 − + 1 at the point whose x-coordinate is 2.


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 (0, 5)


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.]


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.


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 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 slope of the tangent and the normal to the following curve at the indicted point  y = (sin 2x + cot x + 2)2 at x = 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 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 = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?


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 y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?


Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?


Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?


Find the equation of the tangent and the normal to the following curve at the indicated points:

x = 3cosθ − cos3θ, y = 3sinθ − sin3θ? 


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  2y2 = x3 and y2 = 32x ?


Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?


Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?


Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?


Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?


Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?


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 ?


The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .


Find the equation of a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis


Find the angle of intersection of the curves y2 = x and x2 = y.


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 curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a 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 slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is


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 ______.


The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.


The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×