Advertisements
Advertisements
प्रश्न
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Advertisements
उत्तर
Slope of the given tangent is 0.
\[\text { Let }\left( x_1 , y_1 \right)\text { be a point where the tangent is drawn to the curve} (1).\]
\[\text { Since, the point lies on the curve } . \]
\[\text { Hence }, y_1 = \frac{1}{{x_1}^2 - 2 x_1 + 3} . . . \left( 1 \right) \]
\[\text { Now,} y = \frac{1}{x^2 - 2x + 3}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( x^2 - 2x + 3 \right)\left( 0 \right) - \left( 2x - 2 \right)1}{\left( x^2 - 2x + 3 \right)^2} = \frac{- 2x + 2}{\left( x^2 - 2x + 3 \right)^2}\]
\[\text { Slope of tangent }=\frac{- 2 x_1 + 2}{\left( {x_1}^2 - 2 x_1 + 3 \right)^2}\]
\[\text { Given that }\]
\[\text { Slope of tangent = slope of the given line }\]
\[ \Rightarrow \frac{- 2 x_1 + 2}{\left( {x_1}^2 - 2 x_1 + 3 \right)^2} = 0\]
\[ \Rightarrow - 2 x_1 + 2 = 0\]
\[ \Rightarrow 2 x_1 = 2\]
\[ \Rightarrow x_1 = 1\]
\[\text { Now }, y = \frac{1}{1 - 2 + 3} = \frac{1}{2} ............\left[ \text { From }\left( 1 \right) \right]\]
\[ \therefore \left( x_1 , y_1 \right) = \left( 1, \frac{1}{2} \right)\]
\[\text { Equation oftangentis},\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - \frac{1}{2} = 0 \left( x - 1 \right)\]
\[ \Rightarrow y = \frac{1}{2}\]
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 the equation of tangents to the curve y= x3 + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.
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 points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
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 slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
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?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point x2 = 4y at (2, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
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 curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
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 to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y
The curve y = `x^(1/5)` has at (0, 0) ______.
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is:
The line is y = x + 1 is a tangent to the curve y2 = 4x at the point.
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 the curves y2 = 6x, 9x2 + by2 = 16, cut each other at right angles then the value of b 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 ______.
