Advertisements
Advertisements
प्रश्न
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
उत्तर
Equations of the given circles are
`x^2+y^2=4 ................(1)`
`(x-2)^2+y^2=4...........(2)`
Equation (1) is a circle with centre O at the origin and radius 2. Equation (2) is a circle with centre C (2, 0) and radius 2. Solving equations (1) and (2), we have
`(x-2)^2+y^2=x^2+y^2`
`or x^2-4x+4+y^2=x^2+y^2`
or x=1 which gives `y=+-sqrt3`
Thus, the points of intersection of the given circles are `A(1,sqrt3) and A'(1,-sqrt3) ` as shown in the Figure
Required area of the enclosed region OACA′O between circles
= 2 [area of the region ODCAO]
= 2 [area of the region ODAO + area of the region DCAD]
`=2[int_0^1ydx+int_1^2ydx]`
`=2[int_0^1sqrt(4-(x-2)^2)dx+int_1^2sqrt(4-x^2)dx]`
`=2[1/2(x-2)sqrt(4-(x-2)^2)+1/2xx4sin^(-1)((x-2)/2)]_0^1+2[1/2xsqrt(4-x^2)+1/2xx4sin^(-1)(x/2)]_1^2`
`=[(x-2)sqrt(4-(x-2)^2)+4sin^(-1)((x-2)2)]_0^1+[xsqrt(4-x^2)+4sin^(-1)(x/2)]_1^2`
`=[(-sqrt3+4sin^(-1)(-1/2))-4sin^(-1)(-1)]+[4sin^(-1)1-sqrt3-4sin^(-1)(1/2)]`
`=[(-sqrt3-4xxpi/6)+4xxpi/2]+[4xxpi/2-sqrt3-4xxpi/6]`
`=(8pi)/3-2sqrt3`
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
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 all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
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)`
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)
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 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 slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
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 \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] 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 = 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 \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x 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.
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
