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(x/y) + 5x - 7 = - (3/4).y
x + 5xy - 7y = - 3y²/4
4x + 20xy - 28y = - 3y²
3y² - 28y + 20xy + 4x = 0
3y² - 28y + 20xy + 4x = 0 → when x is constant: → (6y - 28 + 20x).dy = 0
3y² - 28y + 20xy + 4x = 0 → when y is constant: → (20y + 4).dx = 0
(6y - 28 + 20x).dy + (20y + 4).dx = 0
(6y - 28 + 20x).dy = - (20y + 4).dx
(3y - 14 + 10x).dy = - (10y + 2).dx
dy/dx = - (10y + 2)/(3y - 14 + 10x) → given point P (1 ; 2)
dy/dx = - (20 + 2)/(6 - 14 + 10)
dy/dx = - 22/2
dy/dx = - 11 ← this is the slope of tha tangent line to the curve at x = 1
The typical equation of a line is: y = mx + y₀ → where m: slope and where y₀: y-intercept
The slope of the tangent line is (- 11), so the equation of the tangent line becomes: y = - 11x + y₀
The tangent line passes through P (1 ; 2), so these coordinates must verify the equation of the tangent line.
y = - 11x + y₀
y₀ = y + 11x → you substitute x and y by the coordinates of the point P (1 ; 2)
y₀ = 2 + 11
y₀ = 13
→ The equation of the tangent line at P is: y = - 11x + 13
Pope
Begin by writing the equation in general form.
x/y + 5x - 7 = -3/4y
4x + 20xy - 28y = -3y² ... (y ≠ 0)
3y² + 20xy + 4x - 28y = 0
Point P does satisfy the equation, which makes our job easier. This is a second-degree equation in x and y. The tangent line must include P, and it would be a bit simpler here to substitute for x, so let this be the line of tangency:
x = k(y - 2) + 1
x = ky + 1 - 2k
3y² + 20xy + 4x - 28y = 0
3y² + 20y(ky + 1 - 2k) + 4(ky + 1 - 2k) - 28y = 0
(3 + 20k)y² + (-8 - 36k)y + (4 - 8k) = 0
It is a quadratic equation in y. Let the discriminant equal zero.
(-8 - 36k)² - 4(3 + 20k)(4 - 8k) = 0
(2 + 9k)² - (3 + 20k)(1 - 2k) = 0
121k² + 22k + 1 = 0
(11k + 1)² = 0
k = -1/11
Substitute that into the tangent line equation.
x = ky + 1 - 2k
x = (-1/11)y + 1 - 2(-1/11)
11x + y - 13 = 0
rotchm
MAny ways here:
Performing implicit differentiation, find y'. this is the slope m.
At P, this slope is thus ...? Thus Y = mX + B
From P, you find B.
Done!
Another way, view itr all as x = x(y) and find x'(y). No implicit differentiation required...
If need be, show your steps and answers here. Then we can take it from there.