Advertisements
Advertisements
प्रश्न
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
Advertisements
उत्तर
Abscissa means the horizontal co-ordiante of a point.
Given that abscissa = 2.
i.e., x = 2
\[x^2 + 2 y^2 - 4x - 6y + 8 = 0 . . . \left( 1 \right)\]
\[\text { Differentiating both sides w.r.t.x }, \]
\[2x + 4y\frac{dy}{dx} - 4 - 6\frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx}\left( 4y - 6 \right) = 4 - 2x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{4 - 2x}{4y - 6} = \frac{2 - x}{2y - 3}\]
\[\text { When }x=2,\text { from } (1), \text { we get }\]
\[4 + 2 y^2 - 8 - 6y + 8 = 0\]
\[ \Rightarrow 2 y^2 - 6y + 4 = 0\]
\[ \Rightarrow y^2 - 3y + 2 = 0\]
\[ \Rightarrow \left( y - 1 \right)\left( y - 2 \right) = 0\]
\[ \Rightarrow y = 1 ory = 2\]
\[\text { Case }-1:y = 1\]
\[\text { Slope of tangent } = \left( \frac{dy}{dx} \right)_\left( 2, 1 \right) =\frac{0}{- 1}=0\]
\[\left( x_1 , y_1 \right) = \left( 2, 1 \right)\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - 1 = \frac{- 1}{0} \left( x - 2 \right)\]
\[ \Rightarrow x - 2 = 0\]
\[ \Rightarrow x = 2\]
\[\text { Case}-2:y = 2\]
\[\text { Slope of tangent} = \left( \frac{dy}{dx} \right)_\left( 2, 2 \right) =\frac{0}{1}=0\]
\[\left( x_1 , y_1 \right) = \left( 2, 2 \right)\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - 2 = \frac{- 1}{0} \left( x - 2 \right)\]
\[ \Rightarrow x - 2 = 0\]
\[ \Rightarrow x = 2\]
In both cases, the equation of normal is x = 2
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
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`
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
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.]
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
(A) (1, 2)
(B) (2, 1)
(C) (1, −2)
(D) (−1, 2)
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
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 x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
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) ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
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(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 _____________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes is
(a) \[\left( 4, \frac{8}{3} \right)\]
(b) \[\left( - 4, \frac{8}{3} \right)\]
(c) \[\left( 4, - \frac{8}{3} \right)\]
(d) none of these
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the equation of the tangent line to the curve `"y" = sqrt(5"x" -3) -5`, which is parallel to the line `4"x" - 2"y" + 5 = 0`.
Find the equation of the normal lines to the curve 3x2 – y2 = 8 which are parallel to the line x + 3y = 4.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
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 Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
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 ______.
