Advertisements
Advertisements
Question
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
Solution
Let (x1, y1) be the required point.
The slope of line y = 3x + 4 is 3.
\[\text { Since, the point lies on the curve } . \]
\[\text { Hence, y }_1 = 2 {x_1}^2 - x_1 + 1\]
\[\text { Now, y } = 2 x^2 - x + 1\]
\[\frac{dy}{dx} = 4x - 1\]
\[\text { Now,} \]
\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =4 x_1 -1\]
\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)= \text { Slope of the given line [Given] }\]
\[ \therefore 4 x_1 - 1 = 3\]
\[ \Rightarrow 4 x_1 = 4\]
\[ \Rightarrow x_1 = 1\]
\[\text { and }\]
\[ y_1 = 2 {x_1}^2 - x_1 + 1 = 2 - 1 + 1 = 2\]
\[\text { Thus, the required point is }\left( 1, 2 \right).\]
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
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 slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
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 ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 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 normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
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 all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
The equation of the normal to the curve y = sinx at (0, 0) is ______.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
The normal at the point (1, 1) on the curve `2y + x^2` = 3 is
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 ______.
The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.
