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 equation of tangents to the curve y= x3 + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.
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 = 3x4 − 4x at x = 4.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
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 normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
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 points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
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 y = 2x2 − 3x − 1 at (1, −2) ?
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 = at2, y = 2at at t = 1 ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
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 ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
Find the angle of intersection of the curves y2 = x and x2 = y.
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 point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
