Advertisements
Advertisements
प्रश्न
Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?
Advertisements
उत्तर
\[y=2 x^2 -3x-1\]
\[\text { Differentiating both sides w.r.t.x,} \]
\[\frac{dy}{dx} = 4x - 3\]
\[\text { Given } \left( x_1 , y_1 \right) = \left( 1, - 2 \right)\]
\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_\left( 1, - 2 \right) =4-3=1\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y + 2 = 1 \left( x - 1 \right)\]
\[ \Rightarrow y + 2 = x - 1\]
\[ \Rightarrow x - y - 3 = 0\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y + 2 = - 1 \left( x - 1 \right)\]
\[ \Rightarrow y + 2 = - x + 1\]
\[ \Rightarrow x + y + 1 = 0\]
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.
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 slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
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 parabola y2 = 4ax at the point (at2, 2at).
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
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 points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to 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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
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 angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
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
Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c2 to interest 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
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
The normal at the point (1, 1) on the curve `2y + x^2` = 3 is
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
