Advertisements
Advertisements
प्रश्न
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1
Advertisements
उत्तर
Given that: ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.
Differentiating both sides w.r.t. x
`"d"/"dx" ("a"x^2 + 2"h"xy + "b"y^2 + 2"g"x + 2"f"y + "c") = "d"/"dx" (0)`
⇒ `"a"*2x + 2"h"(x * "dy"/"dx" + y*1) + "b"*2*y* "dy"/"dx" + 2"g"*1 + 2"f"* "dy"/"dx" + 0` = 0
⇒ `2"a"x + 2"h"x * "dy"/"dx" + 2"h"y + 2"b"y * "dy"/"dx" + 2"g" + 2"f" * "dy"/"dx"` = 0
⇒ `2"h"x * "dy"/"dx" + 2"b"y "dy"/"dx" + 2"f" "dy"/"dx"` = – 2ax – 2hy – 2g
⇒ `(2"h"x + 2"b"y + 2"f") "dy"/"dx"` = – 2(ax + hy + g)
⇒ `2("h"x + "b"y + "f") "dy"/"dx"` = = – 2(ax + hy + g)
⇒ `"dy"/"dx" = (-2("a"x + "h"y + "g"))/(2("h"x + "b"y + "f"))`
⇒ `"dy"/"dx" = (-("a"x + "h"y + "g"))/(("h"x + "b"y + "f"))`
Now, differentiating the given equation w.r.t. y.
`"d"/"dy" ("a"x^2 + 2"h"xy + "b"y^2 + 2"g"x + 2"f"y + "c") = "d"/"dy"(0)`
⇒ `2"a"x* "dx"/"dy" + 2"h" (y * "dx"/"dy" + x*1) + 2"b"y + 2"g" * "dx"/"dy" + 2"f" * 1 + 0` = 0
⇒ `2"a"x * "dx"/"dy" + 2"h"y * "dx"/"dy" + 2"h"x + 2"b"y + 2"g" * "dx"/"dy" + 2"f"` = 0
⇒ `2"a"x "dx"/"dy" + 2"h"y * "dx"/"dy" + 2"g" * "dx"/"dy"` = – 2hx – 2by – 2f
⇒ `(2"a"x + 2"h"y + 2"g") "dx"/"dy"` = = – 2hx – 2by – 2f
⇒ `"dx"/"dy" = (-2"h"x - 2"b"y - 2"f")/(2"a"x + 2"h"y + 2"g")`
⇒ `"dx"/"dy" = (-2("h"x + "b"y + "f"))/(2("a"x + "h"y + "g"))`
⇒ `"dx"/"dy" = (-("h"x + "b"y + "f"))/(("a"x + "h"y + "g"))`
∴ `"dy"/"dx" * "dx"/"dy" = [(-("a"x + "h"y + "g"))/(("h"x + "b"y + "f"))][(-("h"x + "b"y + "f"))/(("a"x + "h"y + "g"))]` = 1
Hence, `"dy"/"dx" * "dx"/"dy"` = 1.
Hence proved.
APPEARS IN
संबंधित प्रश्न
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
x2 + 3x + 2
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
sin (log x)
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If y = Aemx + Benx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`.
Find `("d"^2"y")/"dx"^2`, if y = `"x"^5`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
`sin xy + x/y` = x2 – y
tan–1(x2 + y2) = a
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If x2 + y2 + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is ______.
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.
If x = a cos t and y = b sin t, then find `(d^2y)/(dx^2)`.
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Find `(d^2y)/dx^2, "if" y = e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
Let \[y=f(x)\]. What is the first derivative of \[y\] with respect to \[x\]?
When is the second order derivative of \[y\] with respect to \[x\] defined?
Which expression defines the second order derivative of \[y\] with respect to \[x\]?
Which equation is satisfied by \[y=\mathrm{A}\sin x+\mathrm{B}\cos x\]?
After differentiating \[\sqrt{1-x^2}\cdot\frac{dy}{dx}=1\], which equation results?
What equation is obtained after multiplying through by \[\sqrt{1-x^2}\] in the derivation for \[y=\sin^{-1}x\]?
Which equation follows from \[(1-x^2)\cdot2y_{1}y_{2}+y_{1}^{2}(0-2x)=0\]?
