Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the equation of all lines having slope −1 that are tangents to the curve `y = 1/(x -1), x != 1`
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 points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
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 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
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 point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
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) ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
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 \[\frac{dy}{dx}\] ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
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 the tangent line to the curve `"y" = sqrt(5"x" -3) -5`, which is parallel to the line `4"x" - 2"y" + 5 = 0`.
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
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
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 ______.
