Advertisements
Advertisements
प्रश्न
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
विकल्प
(0, 0)
(2, 16)
(3, 9)
none of these
Advertisements
उत्तर
None of these
\[\text { Let }\left( x_1 , y_1 \right)\text { be the required point.}\]
\[\text { Since, the point lies on the curve }, \]
\[\text { Hence }, y_1 = 12 x_1 - {x_1}^2 \]
\[\text { Now }, \]
\[y = 12x - x^2 \]
\[ \Rightarrow \frac{dy}{dx} = 12 - 2x\]
\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =12 - 2 x_1 \]
\[\text { Given }:\]
\[\text { Slope of the tangent }=0\]
\[12 - 2 x_1 = 0\]
\[ \Rightarrow x_1 = 6\]
\[\text { Now }, \]
\[ y_1 = 12 x_1 - {x_1}^2 = 72 - 36 = 36\]
\[ \therefore \left( x_1 , y_1 \right) = \left( 6, 36 \right)\]
APPEARS IN
संबंधित प्रश्न
Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
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
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
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 y = (sin 2x + cot x + 2)2 at x = π/2 ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
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 = at2, y = 2at at t = 1 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
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 angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
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 angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c2 to interest orthogonally.
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
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?
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
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
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 is y = x + 1 is a tangent to the curve y2 = 4x at the point.
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
An edge of variable cube is increasing at the rate of 3 cm/s. The volume of the cube increasing fast when the edge is 10 cm long is ______ cm3/s.
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 ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
