Advertisements
Advertisements
प्रश्न
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
विकल्प
x + 3y = 8
x + 3y + 8 = 0
x + 3y ± 8 = 0
x + 3y = 0
Advertisements
उत्तर
x + 3y ± 8 = 0
The slope of line x + 3y = 8 is \[\frac{- 1}{3}\].
\[\text { Since the normal is parallel to the given line, the equation of normal will be of the given form.}\]
\[x + 3y = k \]
\[3 x^2 - y^2 = 8 \]
\[\text { Let} \left( x_1 , y_1 \right) \text { be the point of intersection of the two curves} . \]
\[\text { Then }, \]
\[ x_1 + 3 y_1 = k . . . \left( 1 \right)\]
\[3 {x_1}^2 - {y_1}^2 = 8 . . . \left( 2 \right)\]
\[\text { Now, } 3 x^2 - y^2 = 8 \]
\[\text { On differentiating both sides w.r.t.x, we get }\]
\[6x - 2y\frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{6x}{2y} = \frac{3x}{y}\]
\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{3 x_1}{y_1}\]
\[\text { Slope of the normal }, m=\frac{- 1}{\left( \frac{3x}{y} \right)}=\frac{- y_1}{3 x_1}\]
\[\text { Given }:\]
\[\text { Slope of the normal = Slope of the given line }\]
\[ \Rightarrow \frac{- y_1}{3 x_1} = \frac{- 1}{3}\]
\[ \Rightarrow y_1 = x_1 . . . \left( 3 \right)\]
\[\text { From } (2), \text { we get }\]
\[3 {x_1}^2 - {x_1}^2 = 8\]
\[ \Rightarrow 2 {x_1}^2 = 8\]
\[ \Rightarrow {x_1}^2 = 4\]
\[ \Rightarrow x_1 = \pm 2\]
\[\text { Case } 1:\]
\[\text { When } x_1 =2\]
\[\text { From (3), we get }\]
\[ y_1 = x_1 = 2\]
\[ \therefore \left( x_1 , y \right) = \left( 2, 2 \right)\]
\[\text { From (1), we get }\]
\[2 + 3\left( 2 \right) = k\]
\[ \Rightarrow 2 + 6 = k\]
\[ \Rightarrow k = 8\]
\[ \therefore \text { Equation of the normal from} (1)\]
\[ \Rightarrow x + 3y = 8\]
\[ \Rightarrow x + 3y - 8 = 0\]
\[\text { Case } 2:\]
\[\text { When } x_1 =-2\]
\[\text { From (3), we get }\]
\[ y_1 = x_1 = - 2, \]
\[ \therefore \left( x_1 , y \right) = \left( - 2, - 2 \right)\]
\[\text { From (1), we get }\]
\[ - 2 + 3\left( - 2 \right) = k\]
\[ \Rightarrow - 2 - 6 = k\]
\[ \Rightarrow k = - 8\]
\[ \therefore \text { Equation of the normal from } (1)\]
\[ \Rightarrow x + 3y = - 8\]
\[ \Rightarrow x + 3y + 8 = 0\]
\[\text { From both the cases, we get the equation of the normal as }:\]
\[x + 3y \pm 8 = 0\]
APPEARS IN
संबंधित प्रश्न
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
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 x2 + 3y + y2 = 5 at (1, 1) ?
If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
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 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 y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
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 and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
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 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 ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
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 normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
The tangent to the curve y = 2x2 - x + 1 is parallel to the line y = 3x + 9 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.
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 ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.
