Advertisements
Advertisements
Question
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
Options
`(1/2, 1/4)`
`(1/4, 1/2)`
(4, 2)
(1, 1)
Solution
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is `(1/4, 1/2)`.
Explanation:
`"dy"/"dx" = 1/(2y) = tan pi/4` = 1
⇒ y = `1/2`
⇒ x = `1/4`
APPEARS IN
RELATED QUESTIONS
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
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)
Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 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 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 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 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
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 slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
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 \[\frac{dy}{dx}\] ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
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 equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
If the straight line x cosα + y sinα = p touches the curve `x^2/"a"^2 + y^2/"b"^2` = 1, then prove that a2 cos2α + b2 sin2α = p2.
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.
