Advertisements
Advertisements
प्रश्न
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
Advertisements
उत्तर
Equation of curve is given by `sqrt(x) + sqrt(y)` = 4
Let (x1, y1) be the required point on the curve
∴ `sqrt(x)_1 + sqrt(y)_1` = 4
Differentiating both sides w.r.t. x1, we get
`"d"/("dx"_1) sqrt(x_1) + "d"/("dx"_1) sqrt(y_1) = "d"/("dx"_1) (4)`
⇒ `1/(2sqrt(x_1)) + 1/(2sqrt(y_1)) * ("d"y_1)/("dx"_1)` = 0
⇒ `1/sqrt(x_1) + 1/sqrt(y_1) * ("dy"_1)/("dx"_1)` = 0
⇒ `("dy"_1)/("d"x_1) = - sqrt(y_1)/sqrt(x_1)` .....(i)
Since the tangent to the given curve at (x1, y1) is equally inclined to the axes.
∴ Slope of the tangent `("dy"_1)/("dx"_1) = +- tan pi/4` = ±1
So, from equation (i) we get
`- sqrt(y_1)/sqrt(x_1)` = ±1
Squaring both sides, we get
`(y_1)/(x_1)` = 1
⇒ y1 = x1
Putting the value of y1 in the given equation of the curve.
`sqrt(x_1) + sqrt(y_1)` = 4
⇒ `sqrt(x_1) + sqrt(x_1)` = 4
⇒ `2sqrt(x_1)` = 4
⇒ `sqrt(x_1)` = 2
⇒ x1 = 4
Since y1 = x1
∴ y1 = 4
Hence, the required point is (4, 4).
APPEARS IN
संबंधित प्रश्न
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)
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`
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
(A) 3
(B) 1/3
(C) −3
(D) `-1/3`
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
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 points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
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 angle between the curves y = e−x and y = ex at their point of intersections ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
The equation of the normal to the curve y = sinx at (0, 0) is ______.
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Find the equation of the normal lines to the curve 3x2 – y2 = 8 which are parallel to the line x + 3y = 4.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
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
