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7 The equation of a curve is 3x2+4xy+3y2=53 x^{2}+4 x y+3 y^{2}=5. (a) Show that $\frac{\m...
Mar 15, 2024
7 The equation of a curve is 3x2+4xy+3y2=53 x^{2}+4 x y+3 y^{2}=5. (a) Show that dy dx=−3x+2y2x+3y\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3 x+2 y}{2 x+3 y}. (b) Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to y+2x=0y+2 x=0.
Solution by Steps
step 1
To show that dydx=−3x+2y2x+3y\frac{dy}{dx} = -\frac{3x + 2y}{2x + 3y}, we differentiate the given equation implicitly with respect to x
step 2
Differentiating 3x2+4xy+3y2=53x^2 + 4xy + 3y^2 = 5 gives us 6x+4y+4xdydx+6ydydx=06x + 4y + 4x\frac{dy}{dx} + 6y\frac{dy}{dx} = 0
step 3
Rearrange the terms to isolate dydx\frac{dy}{dx}, resulting in dydx(4x+6y)=−6x−4y\frac{dy}{dx}(4x + 6y) = -6x - 4y
step 4
Divide both sides by 4x+6y4x + 6y to get dydx=−6x+4y4x+6y\frac{dy}{dx} = -\frac{6x + 4y}{4x + 6y}, which simplifies to dydx=−3x+2y2x+3y\frac{dy}{dx} = -\frac{3x + 2y}{2x + 3y}
Answer
dydx=−3x+2y2x+3y\frac{dy}{dx} = -\frac{3x + 2y}{2x + 3y}
Key Concept
Implicit Differentiation
Explanation
To find the derivative of y with respect to x for an equation involving both x and y, we use implicit differentiation and then solve for dydx\frac{dy}{dx}.
step 5
To find the points where the tangent is parallel to y+2x=0y + 2x = 0, we set dydx\frac{dy}{dx} equal to the slope of this line, which is -1
step 6
Substitute y=−2xy = -2x into the original equation to find the x-coordinates of the points
step 7
Substituting gives us 3x2+4x(−2x)+3(−2x)2=53x^2 + 4x(-2x) + 3(-2x)^2 = 5
step 8
Simplifying the equation results in 3x2−8x2+12x2=53x^2 - 8x^2 + 12x^2 = 5, which simplifies to 7x2=57x^2 = 5
step 9
Solving for x gives us x2=57x^2 = \frac{5}{7}, so x=±57x = \pm\sqrt{\frac{5}{7}}
step 10
Using y=−2xy = -2x, we find the y-coordinates: y=−2(±57)y = -2(\pm\sqrt{\frac{5}{7}}), which gives us y=∓207y = \mp\sqrt{\frac{20}{7}}
Answer
The exact coordinates are (57,−257)\left(\sqrt{\frac{5}{7}}, -2\sqrt{\frac{5}{7}}\right) and (−57,257)\left(-\sqrt{\frac{5}{7}}, 2\sqrt{\frac{5}{7}}\right).
Key Concept
Parallel Tangents and Implicit Differentiation
Explanation
To find where the tangent to a curve is parallel to a given line, we set the derivative equal to the slope of the line and solve for the coordinates.
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