Advertisements
Advertisements
प्रश्न
If the line y = 4x – 5 touches the curves y2 = ax3 + b at the point (2, 3), find a and b.
उत्तर
y2 = ax3 + b
Differentiating both sides w.r.t. x, we get
`2ydy/dx = a xx 3x^2 + 0`
∴ `dy/dx = (3ax^2)/(2y)`
∴ `(dy/dx)_("at" (2, 3)) = (3a(2)^2)/(2(3))`
= 2a
= slope of the tangent at (2, 3)
Since, the line y = 4x – 5 touches the curve at the point (2, 3), slope of the tangent at (2, 3) is 4.
∴ 2a = 4
∴ a = 2
Since (2, 3) lies on the curve y2 = ax3 + b,
(3)2 = a(2)3 + b
∴ 9 = 8a + b
∴ 9 = 8(2) + b ...[∵ a = 2]
∴ b = – 7
Hence, a = 2 and b = – 7.
APPEARS IN
संबंधित प्रश्न
Find the equation of tangent and normal to the curve at the point on it.
y = x2 + 2ex + 2 at (0, 4)
Find the equations of tangents and normals to the following curves at the indicated points on them: `x^2 - sqrt(3)xy + 2y^2 = 5 "at" (sqrt(3),2)`
Find the equations of tangents and normals to the following curves at the indicated points on them : x sin 2y = y cos 2x at `(pi/4, pi/2)`
Find the equations of tangents and normals to the following curve at the indicated points on them:
x = sin θ and y = cos 2θ at θ = `pi/(6)`
Find the equations of tangents and normals to the following curves at the indicated points on them : `x = sqrt(t), y = t - (1)/sqrt(t)` at = 4.
Find the points on the curve y = x3 – 2x2 – x where the tangents are parllel to 3x – y + 1 = 0.
Find the equation of the tangents to the curve x2 + y2 – 2x – 4y + 1 =0 which a parallel to the X-axis.
Find the equations of the normals to the curve 3x2 – y2 = 8, which are parallel to the line x + 3y = 4.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.
If each side of an equilateral triangle increases at the rate of `(sqrt(2)"cm")/sec`, find the rate of increase of its area when its side of length 3 cm.
If water is poured into an inverted hollow cone whose semi-vertical angle is 30°, so that its depth (measured along the axis) increases at the rate of`( 1"cm")/sec`. Find the rate at which the volume of water increasing when the depth is 2 cm.
Choose the correct option from the given alternatives :
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.
Choose the correct option from the given alternatives :
The normal to the curve x2 + 2xy – 3y2 = 0 at (1, 1)
Choose the correct option from the given alternatives :
The equation of the tangent to the curve y = `1 - e^(x/2)` at the point of intersection with Y-axis is
Choose the correct option from the given alternatives :
If the tangent at (1, 1) on y2 = x(2 – x)2 meets the curve again at P, then P is
Solve the following : If the curves ax2 + by2 = 1 and a'x2 + b'y2 = 1, intersect orthogonally, then prove that `(1)/a - (1)/b = (1)/a' - (1)/b'`.
Solve the following : Determine the area of the triangle formed by the tangent to the graph of the function y = 3 – x2 drawn at the point (1, 2) and the coordinate axes.
The slope of the tangent to the curve x = 2 sin3θ, y = 3 cos3θ at θ = `pi/4` is ______.
The slope of the normal to the curve y = x2 + 2ex + 2 at (0, 4) is ______.
If the line y = 4x – 5 touches the curve y2 = ax3 + b at the point (2, 3) then a + b is
If the tangent at (1, 1) on y2 = x(2 − x)2 meets the curve again at P, then P is
Find the slope of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`
Find the slope of normal to the curve 3x2 − y2 = 8 at the point (2, 2)
Find the slope of tangent to the curve x = sin θ and y = cos 2θ at θ = `pi/6`
Find the equation of normal to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`
Find points on the curve given by y = x3 − 6x2 + x + 3, where the tangents are parallel to the line y = x + 5.
Find the equation of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`.
Find the equation of the tangents to the curve x2 + y2 – 2x – 4y + 1 = 0 which is parallel to the X-axis.
The coordinates of the point on the curve y = 2x – x2, the tangent at which has slope 4 are ______.
Find the points on the curve y = x3 – 9x2 + 15x + 3 at which the tangents are parallel to y-axis.
If the line 2x – y + 7 = 0 touches the curve y = ax2 + bx + 5 at (1, 9) then the values of ‘a’ and ‘b’ are ______.
