Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Advertisements
उत्तर
\[4 x^2 + 9 y^2 = 36\]
\[\text { Differentiating both sides w.r.t.x }, \]
\[8x + 18y \frac{dy}{dx} = 0\]
\[ \Rightarrow 18y \frac{dy}{dx} = - 8x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- 8x}{18y} = \frac{- 4x}{9y}\]
\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_\left( 3 \cos\theta, 2 \sin\theta \right) =\frac{- 12\cos\theta}{18\sin\theta}=\frac{- 2 \cos\theta}{3 \sin\theta}\]
\[\text { Given} \left( x_1 , y_1 \right) = \left( 3 \cos\theta, 2 \sin\theta \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 2 \sin\theta = \frac{- 2 \cos\theta}{3 \sin\theta}\left( x - 3 \cos\theta \right)\]
\[ \Rightarrow 3y \sin\theta - 6 \sin^2 \theta = - 2x \cos\theta + 6 \cos^2 \theta\]
\[ \Rightarrow 2x \cos\theta + 3y \sin\theta = 6\left( \cos^2 \theta + \sin^2 \theta \right)\]
\[ \Rightarrow 2x \cos\theta + 3y \sin\theta = 6\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - 2 \sin\theta = \frac{3 \sin\theta}{2 \cos\theta}\left( x - 3 \cos\theta \right)\]
\[ \Rightarrow 2y \cos\theta - 4 \sin\theta \cos\theta = 3x \sin\theta - 9 \sin\theta \cos\theta\]
\[ \Rightarrow 3x \sin\theta - 2y \cos\theta - 5\sin\theta \cos\theta = 0\]
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 points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t = `pi/4`
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 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 x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
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 ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 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 equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
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 equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
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 at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
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 _____________ .
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
The curve y = `x^(1/5)` has at (0, 0) ______.
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
The tangent to the curve y = 2x2 - x + 1 is parallel to the line y = 3x + 9 at the point ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
Which of the following represent the slope of normal?
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane 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 ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
