Advertisements
Advertisements
प्रश्न
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
विकल्प
x + y = 0
x − y = 0
x + y + 1 = 0
x − y = 1
Advertisements
उत्तर
`x − y = 0`
\[\text { Given }: \]
\[2y + x^2 = 3\]
\[ \Rightarrow 2\frac{dy}{dx} + 2x = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- 2x}{2} = - x\]
\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) =-1\]
\[\text { Slope of the normal },m=\frac{- 1}{\text { Slope of the tangent }}=\frac{- 1}{- 1}=1\]
\[\text { Now }, \]
\[\left( x_1 , y_1 \right) = \left( 1, 1 \right)\]
\[ \therefore \text { Equation of the normal }\]
\[ = y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 1 = 1 \left( x - 1 \right)\]
\[ \Rightarrow x - y = 0\]
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find the equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
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 equation of the normal to curve y2 = 4x at the point (1, 2).
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to 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 = x2 at (0, 0) ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
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` = ____________.
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
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 points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of 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 ______.
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 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 ______.
