Advertisements
Advertisements
प्रश्न
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Advertisements
उत्तर
Slope of x - axis is 0
Let (x1, y1) be the required point.
\[y = 2 x^3 - 15 x^2 + 36x - 21\]
\[\text { Since }\left( x_1 , y_1 \right) \text { lies on the curve . Therefore } \]
\[ y_1 = 2 {x_1}^3 - 15 {x_1}^2 + 36 x_1 - 21 . . . \left( 1 \right)\]
\[\text { Now,} y = 2 x^3 - 15 x^2 + 36x - 21\]
\[ \Rightarrow \frac{dy}{dx} = 6 x^2 - 30x + 36\]
\[\text { Slope of tangent at }\left( x_1 , y_1 \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = 6 {x_1}^2 - 30 x_1 + 36\]
\[\text { Given that }\]
\[\text { Slope of tangent at }\left( x, y \right)= \text { slope of thex-axis }\]
\[6 {x_1}^2 - 30 x_1 + 36 = 0\]
\[ \Rightarrow {x_1}^2 - 5 x_1 + 6 = 0\]
\[ \Rightarrow \left( x_1 - 2 \right)\left( x_1 - 3 \right) = 0\]
\[ \Rightarrow x_1 = 2 \text{ or }x_1 = 3\]
\[\text { Case }1: x_1 = 2\]
\[ y_1 = 16 - 60 + 72 - 21 = 7 ...............[\text { From } (1)]\]
\[\left( x_1 , y_1 \right) = \left( 2, 7 \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 7 = 0\left( x - 2 \right)\]
\[ \Rightarrow y = 7\]
\[\text { Case }2: x_1 = 3\]
\[ y_1 = 54 - 135 + 108 - 21 = 6 .................[\text { From }(1)]\]
\[\left( x_1 , y_1 \right) = \left( 3, 6 \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 6 = 0\left( x - 3 \right)\]
\[ \Rightarrow y = 6\]
APPEARS IN
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 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
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
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 = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
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?
The curve y = `x^(1/5)` has at (0, 0) ______.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
The tangent to the curve y = 2x2 - x + 1 is parallel to the line y = 3x + 9 at the point ____________.
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
