Advertisements
Advertisements
प्रश्न
Using integration find the area of the triangle formed by negative x-axis and tangent and normal to the circle `"x"^2 + "y"^2 = 9 "at" (-1,2sqrt2)`.
Advertisements
उत्तर
The equation of circle is `"x"^2 + "y"^2 = 9`
∴ `2"x" + 2"y"(d"y")/(d"x") = 0`
⇒ `(d"y")/(d"x") = -("x")/("y")`
Slope of tangent at `(-1,2sqrt(2)) and "m"_"T" = (1)/(2sqrt(2)`
∴ eq. of tangent : `"y" - 2sqrt(2) = (1)/(2sqrt(2))("x"+ 1)`
⇒ `"x" - 2sqrt(2) "y" + 9 = 0`
Clearly it cuts x axis at `(-9,0)`
Also eq. of normal : `"y"-2sqrt(2) = -2sqrt(2)("x"+1)`
⇒ `2sqrt(2"x")+"y" = 0`
As tangent and normal both meet at the point `(-1,2sqrt(2))`.
So, ar(OPB) = `int_-9^-1 ("x"+9)/(2sqrt(2))d"x" + int_-1^0 -2sqrt(2)"x"d"x"`
⇒ = `(1)/(2sqrt(2))["x"^2/(2) + 9"x"]_-9^-1 - sqrt2["x"^2/2]_-1^0`
⇒ = `(1)/(2sqrt2)[(1/2 - 9)-(81/2 - 81)]-sqrt(2)[0-1]`
∴ ar(OPB) = `9sqrt2 "sq.units"`.
APPEARS IN
संबंधित प्रश्न
Find the area of the region in the first quadrant enclosed by x-axis, line x = `sqrt3` y and the circle x2 + y2 = 4.
The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.
Find the area under the given curve and given line:
y = x2, x = 1, x = 2 and x-axis
Find the area under the given curve and given line:
y = x4, x = 1, x = 5 and x-axis
Find the area of the smaller region bounded by the ellipse `x^2/a^2 + y^2/b^2 = 1` and the line `x/a + y/b = 1`
Find the area of the smaller region bounded by the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1\] and the line \[\frac{x}{3} + \frac{y}{2} = 1 .\]
Find the area of the region bounded by the following curves, the X-axis and the given lines: 2y = 5x + 7, x = 2, x = 8
Choose the correct alternative :
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _____.
Using definite integration, area of the circle x2 + y2 = 49 is _______.
State whether the following is True or False :
The area bounded by the two cures y = f(x), y = g (x) and X-axis is `|int_"a"^"b" f(x)*dx - int_"b"^"a" "g"(x)*dx|`.
Choose the correct alternative:
Area of the region bounded by the curve y = x3, x = 1, x = 4 and the X-axis is ______
Choose the correct alternative:
Area of the region bounded by x = y4, y = 1 and y = 5 and the Y-axis lying in the first quadrant is ______
State whether the following statement is True or False:
The area bounded by the curve y = f(x) lies on the both sides of the X-axis is `|int_"a"^"b" "f"(x) "d"x| + |int_"b"^"c" "f"(x) "d"x|`
The area of the region bounded by the curve y2 = 4x, the X axis and the lines x = 1 and x = 4 is ______
The area of the region bounded by the curve y2 = x and the Y axis in the first quadrant and lines y = 3 and y = 9 is ______
The area of the region x2 = 4y, y = 1 and y = 2 and the Y axis lying in the first quadrant is ______
Find the area of the region bounded by the parabola y2 = 25x and the line x = 5
Find the area of the region bounded by the curve y = `sqrt(9 - x^2)`, X-axis and lines x = 0 and x = 3
Find the area of the region bounded by the curve y = `sqrt(2x + 3)`, the X axis and the lines x = 0 and x = 2
Find area of the region bounded by the curve y = – 4x, the X-axis and the lines x = – 1 and x = 2
Find area of the region bounded by the parabola x2 = 4y, the Y-axis lying in the first quadrant and the lines y = 3
Find the area of the region bounded by the curve y = `sqrt(36 - x^2)`, the X-axis lying in the first quadrant and the lines x = 0 and x = 6
Area bounded by the curve xy = 4, X-axis between x = 1, x = 5 is ______.
The area bounded by the X-axis, the curve y = f(x) and the lines x = 1, x = b is equal to `sqrt("b"^2 + 1) - sqrt(2)` for all b > 1, then f(x) is ______.
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x, is:
The area included between the parabolas y2 = 4a(x +a) and y2 = 4b(x – a), b > a > 0, is
If area of the region bounded by y ≥ cot( cot–1|In|e|x|) and x2 + y2 – 6 |x| – 6|y| + 9 ≤ 0, is λπ, then λ is ______.
The area bounded by the x-axis and the curve y = 4x – x2 – 3 is ______.
Area bounded by y = sec2x, x = `π/6`, x = `π/3` and x-axis is ______.
The area (in sq. units) of the region {(x, y) : y2 ≥ 2x and x2 + y2 ≤ 4x, x ≥ 0, y ≥ 0} is ______.
