Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Slopes of the tangent line?

Determine the exact coordinates for the four points that are the x andyintercepts of the graph of the equation 10x^2+ 5y^2+ 4xy= 110. Find the slopes of the tangent lines for each of the points.

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10x^2+ 5y^2+ 4xy= 110

for the x-intercept, y = 0

10x^2 = 110

x = +/- sqrt(11)

for the y-intercept. x = 0

5y^2 = 110.

y = +/- sqrt(22)

======================================================

10x^2+ 5y^2+ 4xy = 110

20dx + 10ydy + 4(xdy + ydx) = 0

20dx + 4ydx + 10ydy + 4xdy = 0.

(20 + 4y)dx = -(10y + 4x)dy

dy/dx = -(20 + 4y)/(10y + 4x)

at (0,sqrt(22)) ... m = -(20sqrt(22) + 4)/(10sqrt(22))

. . . . m = -2(sqrt(22) + 4)/(sqrt(22))

==== this isn't fun anymore, just a bunch of algebra involving sq roots, === so since it is your assignment, you can finish.

• ted s
Lv 7
1 month agoReport

' david ' missed an x in the 2nd line...20 x dx

• 10x²+ 5y²+ 4xy= 110

To find x-intercepts, put y=0

10x²+ 5(0)²+ 4x(0)= 110

10x² = 110

x² = 11

x = ±√11

x-intercepts are (√11,0) and (-√11,0)

To find y-intercepts, put x=0

10(0)²+ 5y²+ 4(0)y= 110

5y² = 110

y² = 22

y = ±√22

y-intercepts are (0,√22) and (0,-√22)

To find slope of tangent, find derivative of 10x²+ 5y²+ 4xy= 110

20x+ 10y dy/dx+ 4(y + x dy/dx) = 0

20x+ 10y dy/dx+ 4y + 4x dy/dx = 0

dy/dx(10y+4x) = -20x-4y

dy/dx = -4(5x+y)/2(5y+2x)

dy/dx = -2(5x+y)/(5y+2x)

At (√11,0), slope dy/dx = -2(5√11+0)/(5*0+2√11) = -(10√11)/(2√11) = -5

Using the formula y-y₁=m(x-x₁)

equation of tangent is y-0 = -5(x-√11)

y = -5x + 5√11

At (-√11,0), slope dy/dx = -2(5*-√11+0)/(5*0+2*-√11) = (10√11)/(-2√11) = -5

equation of tangent is y-0 = -5(x-(-√11))

y = -5x - 5√11

At (0,√22), slope dy/dx = -2(5*0+√22)/(5*√22 +2*0) = -(2√22)/(5√22) = -2/5

equation of tangent is y-√22 = (-2/5)(x-0)

y = -(2/5)x + √22

At (0,-√22), slope dy/dx = -2(5*0-√22)/(5*-√22 +2*0) = (2√22)/(-5√22) = -2/5

equation of tangent is y-(-√22) = (-2/5)(x-0)

y = -(2/5)x - √22

If you want to see how the function and tangents look then click on below link

https://www.desmos.com/calculator/j0tzfh3apy

• When x = 0, you have 5y^2 = 110, so

y = +/- sqrt(22), around 4.7.

When y = 0, you have 10x^2 = 110, so

x = +/- sqrt(11), around 3.3.

Implicit differentiation:

20x + 25y dy/dx + 4y + 4x dy/dx = 0.

When x = 0, dy/dx = -4y/25y = -0.16 at both of the y-intercepts.

When y = 0, dy/dx = -20x/4x = -5 at both of the x-intercepts.