Advertisements
Advertisements
Question
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Advertisements
Solution
\[y^2 =4ax\]
\[\text { Differentiating both sides w.r.t.x}, \]
\[2y\frac{dy}{dx} = 4a\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2a}{y}\]
\[\text { At }\left( x_1 , y_1 \right)\]
\[\text { Slope of tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{2a}{y_1}=m\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{2a\left( x - x_1 \right)}{y_1}\]
\[ \Rightarrow y y_1 - {y_1}^2 = 2ax - 2a x_1 \]
\[ \Rightarrow y y_1 - 4a x_1 = 2ax - 2a x_1 \]
\[ \Rightarrow y y_1 = 2ax + 2a x_1 \]
\[ \Rightarrow y y_1 = 2a\left( x + x_1 \right)\]
\[\text { Equation of normal is,}\]
\[y - y_1 = \frac{1}{\text { Slope of tangen}t} \left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{- 1}{m}\left( x - x_1 \right)\]
\[ \Rightarrow y - y_1 = \frac{- y_1}{2a}\left( x - x_1 \right)\]
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
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 at (0, 0) ?
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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
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 equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
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 ?
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 normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
Find the angle of intersection of the curves y2 = x and x2 = y.
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
Which of the following represent the slope of normal?
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.
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 normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.
