Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
Advertisements
उत्तर
\[{xy=c}^2 \]
\[\text { Differentiating both sides w.r.t.x }, \]
\[x\frac{dy}{dx} + y = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- y}{x}\]
\[\text { Given } \left( x_1 , y_1 \right) = \left( ct, \frac{c}{t} \right)\]
\[\text { Slope of tangent,}m= \left( \frac{dy}{dx} \right)_\left( ct, \frac{c}{t} \right) =\frac{- \frac{c}{t}}{ct}=\frac{- 1}{t^2}\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - \frac{c}{t} = \frac{- 1}{t^2} \left( x - ct \right)\]
\[ \Rightarrow \frac{yt - c}{t} = \frac{- 1}{t^2} \left( x - ct \right)\]
\[ \Rightarrow y t^2 - ct = - x + ct\]
\[ \Rightarrow x + y t^2 = 2ct\]
\[\text{ Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - \frac{c}{t} = t^2 \left( x - ct \right)\]
\[ \Rightarrow yt - c = t^3 x - c t^4 \]
\[ \Rightarrow x t^3 - yt = c t^4 - c\]
APPEARS IN
संबंधित प्रश्न
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the slope of the normal to the curve x = acos3θ, y = asin3θ at `theta = pi/4`
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Find the equation of the normal to curve y2 = 4x at the point (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 x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
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 y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
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 = a(θ + sinθ), y = a(1 − cosθ) at θ ?
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 the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
Find the equation of a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
The equation of the normal to the curve y = sinx at (0, 0) is ______.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
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).
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
The normal at the point (1, 1) on the curve `2y + x^2` = 3 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 ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
