Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
Advertisements
उत्तर
\[y^2 =4ax\]
\[\text { Differentiating both sides w.r.t.x,} \]
\[2y \frac{dy}{dx} = 4a\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2a}{y}\]
\[\text { Given } \left( x_1 , y_1 \right) = \left( \frac{a}{m^2}, \frac{2a}{m} \right)\]
\[\text { Slope of tangent }= \left( \frac{dy}{dx} \right)_\left( \frac{a}{m^2}, \frac{2a}{m} \right) =\frac{2a}{\left( \frac{2a}{m} \right)}=m\]
\[\text { Equation of tangent is, }\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - \frac{2a}{m} = m \left( x - \frac{a}{m^2} \right)\]
\[ \Rightarrow \frac{my - 2a}{m} = m\left( \frac{m^2 x - a}{m^2} \right)\]
\[ \Rightarrow my - 2a = m^2 x - a\]
\[ \Rightarrow m^2 x - my + a = 0\]
\[\text { Equation of normal is},\]
\[y - y_1 = \frac{1}{\text { Slope of tangent}} \left( x - x_1 \right)\]
\[ \Rightarrow y - \frac{2a}{m} = \frac{- 1}{m}\left( x - \frac{a}{m^2} \right)\]
\[ \Rightarrow \frac{my - 2a}{m} = \frac{- 1}{m}\left( \frac{m^2 x - a}{m^2} \right)\]
\[ \Rightarrow m^3 y - 2a m^2 = - m^2 x + a\]
\[ \Rightarrow m^2 x + m^3 y - 2a m^2 - a = 0\]
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.
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
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 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`
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 1/2
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 slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
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 = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
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 ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
Which of the following represent the slope of normal?
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
The normal at the point (1, 1) on the curve `2y + x^2` = 3 is
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 ______.
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
