Advertisements
Advertisements
प्रश्न
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
उत्तर
Given that y2 = 4ax .....(i) and x2 = 4by .....(ii)
Solving (i) and (ii), we get
`(x^2/(4"b"))^2` = 4ax
⇒ x4 = 64 ab2x
or x(x3 – 64 ab2) = 0
⇒ x = 0, x = `4"a"^(1/3) "b"^(2/3)`
Therefore, the points of intersection are (0, 0) and `(4"a"^(1/3) "b"^(2/3), 4"a"^(2/3)"b"^(1/3))`.
Again, y2 = 4ax
⇒ `"dy"/"dx" = (4"a")/"dx" = (2"a")/y` and x2 = 4by
⇒ `"dy"/"dx" = (2x)/(4"b") = x/(2"b")`
Therefore, at (0, 0) the tangent to the curve y2 = 4ax is parallel to y-axis and tangent to the curve x2 = 4by is parallel to x-axis.
⇒ Angle between curves = `pi/2`
At `(4"a"^(1/3)"b"^(2/3), 4"a"^(2/3)"b"^(1/3))`, m1 ......(Slope of the tangent to the curve (i))
= `2("a"/"b")^(1/3)`
= `(2"a")/(4"a"^(2/3)"b"^(1/3))`
= `1/2("a"/"b")^(1/3)`, m2 ....(Slope of the tangent to the curve (ii))
= `(4"a"^(1/3)"b"^(2/3))/(2"b")`
= `2("a"/"b")^(1/3)`
Therefore, tan θ = `|("m"_2 - "m"_3)/(1 + "m"_1 "m"_2)|`
= `|(2("a"/"b")^(1/3) - 1/2("a"/"b")^(1/3))/(1 + 2("a"/"b")^(1/3) 1/2("a"/"b")^(1/3))|`
= `(3"a"^(1/3) . "b"^(1/3))/(2("a"^(2/3) + "b"^(2/3))`
Hence, θ = `tan^-1((3"a"^(1/3) . "b"^(1/3))/(2("a"^(2/3) + "b"^(2/3))))`
APPEARS IN
संबंधित प्रश्न
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
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 y2 = 2x3 at which the slope of the tangent is 3 ?
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 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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
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 ______.
At (0, 0) the curve y = x3 + x
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
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.
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
Which of the following represent the slope of normal?
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.
