Advertisements
Advertisements
प्रश्न
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
पर्याय
x − 2y = 2
x − 2y + 2 = 0
2x + y = 4
2x + y − 4 = 0
उत्तर
x − 2y = 2
\[\text { Here }, \]
\[y = x\left( 2 - x \right) = 2x - x^2 \]
\[ \Rightarrow \frac{dy}{dx} = 2 - 2x\]
\[\text { Slope of the tangent }= \left( \frac{dy}{dx} \right)_\left( 2, 0 \right) = 2 - 4 = - 2\]
\[\text { Slope of the normal }, m=\frac{- 1}{- 2}=\frac{1}{2}\]
\[\text { Given }: \]
\[\left( x_1 , y_1 \right) = \left( 2, 0 \right)\]
\[ \therefore \text { Equation of the normal }\]
\[ = y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 0 = \frac{1}{2}\left( x - 2 \right)\]
\[ \Rightarrow 2y = x - 2\]
\[ \Rightarrow x - 2y = 2\]
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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
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)`
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
(A) (1, 2)
(B) (2, 1)
(C) (1, −2)
(D) (−1, 2)
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 xy = 6 at (1, 6) ?
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
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 \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] 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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
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 2y2 = x3 and y2 = 32x ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
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 angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.
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.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
Let `y = f(x)` be the equation of the curve, then equation of normal is
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 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 ______.
