Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Advertisements
उत्तर
\[y^2 =4ax\]
\[\text { Differentiating both sides w.r.t.x}, \]
\[2y\frac{dy}{dx} = 4a\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2a}{y}\]
\[\text { At }\left( x_1 , y_1 \right)\]
\[\text { Slope of tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{2a}{y_1}=m\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{2a\left( x - x_1 \right)}{y_1}\]
\[ \Rightarrow y y_1 - {y_1}^2 = 2ax - 2a x_1 \]
\[ \Rightarrow y y_1 - 4a x_1 = 2ax - 2a x_1 \]
\[ \Rightarrow y y_1 = 2ax + 2a x_1 \]
\[ \Rightarrow y y_1 = 2a\left( x + x_1 \right)\]
\[\text { Equation of normal is,}\]
\[y - y_1 = \frac{1}{\text { Slope of tangen}t} \left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{- 1}{m}\left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{- y_1}{2a}\left( x - x_1 \right)\]
APPEARS IN
संबंधित प्रश्न
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
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.]
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
(A) 3
(B) 1/3
(C) −3
(D) `-1/3`
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 \[y = \sqrt{x} \text { at }x = 9\] ?
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 = 2x2 − 3x − 1 at (1, −2) ?
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 = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
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 equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
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 normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
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 ______.
