Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
उत्तर
\[y^2 =\frac{x^3}{4 - x}\]
\[\text { Differentiating both sides w.r.t.x}, \]
\[2y \frac{dy}{dx} = \frac{\left( 4 - x \right)\left( 3 x^2 \right) - x^3 \left( - 1 \right)}{\left( 4 - x \right)^2} = \frac{12 x^2 - 3 x^3 + x^3}{\left( 4 - x \right)^2} = \frac{12 x^2 - 2 x^3}{\left( 4 - x \right)^2}\]
\[\frac{dy}{dx} = \frac{12 x^2 - 2 x^3}{2y \left( 4 - x \right)^2}\]
\[\text { Given } \left( x_1 , y_1 \right) = \left( 2, - 2 \right)\]
\[\text { Slope of tangent,}m= \left( \frac{dy}{dx} \right)_\left( 2, - 2 \right) =\frac{48 - 16}{- 16}=-2\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y + 2 = - 2 \left( x - 2 \right)\]
\[ \Rightarrow y + 2 = - 2x + 4\]
\[ \Rightarrow 2x + y - 2 = 0\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y + 2 = \frac{1}{2} \left( x - 2 \right)\]
\[ \Rightarrow 2y + 4 = x - 2\]
\[ \Rightarrow x - 2y - 6 = 0\]
APPEARS IN
संबंधित प्रश्न
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
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)`
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 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 points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
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 = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
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 y2 = 4x at (1, 2) ?
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 \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
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 ______.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
The equation of normal to the curve y = tanx at (0, 0) is ______.
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
Let `y = f(x)` be the equation of the curve, then equation of normal 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.
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane 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 ______.
