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 equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
Find the slope of the normal to the curve x = acos3θ, y = asin3θ at `theta = pi/4`
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
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.]
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)`
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
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 \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 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.
Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
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 line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The line is y = x + 1 is a tangent to the curve y2 = 4x at the point.
If (a, b), (c, d) are points on the curve 9y2 = x3 where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
