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.
Advertisements
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
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
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 points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
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 a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
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) ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
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 \[\frac{dy}{dx}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
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 normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
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`.
Find the angle of intersection of the curves y2 = x and x2 = y.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
