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 = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
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 equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
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 equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
Find the slope of the tangent and the normal to the following curve at the indicted point x2 + 3y + y2 = 5 at (1, 1) ?
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 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 ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
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 = _____________ .
Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c2 to interest orthogonally.
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
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 normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
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 ____________.
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
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
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
The normal at the point (1, 1) on the curve `2y + x^2` = 3 is
