Math  /  Geometry

QuestionFind all points on the circle x2+y2=676x^{2}+y^{2}=676 where the slope is 512\frac{5}{12} smaller yy-value (x,y)=\quad(x, y)= \square ) larger yy-value (x,y)=((x, y)=(\square \square )

Studdy Solution

STEP 1

1. The given equation x2+y2=676x^2 + y^2 = 676 represents a circle centered at the origin with radius 26.
2. The slope of the tangent to the circle at any point (x,y)(x, y) is given by the derivative dydx\frac{dy}{dx}.
3. The slope of the tangent is specified to be 512\frac{5}{12}.

STEP 2

1. Find the derivative of the circle equation to determine the slope of the tangent at any point (x,y)(x, y).
2. Set the slope of the tangent equal to 512\frac{5}{12} and solve for yy in terms of xx.
3. Substitute the expression for yy back into the circle equation to find the coordinates (x,y)(x, y).
4. Identify the points with smaller and larger yy-values from the solutions.

STEP 3

Differentiate the equation of the circle implicitly with respect to xx to find dydx\frac{dy}{dx}.
Starting with: x2+y2=676 x^2 + y^2 = 676
Differentiating both sides with respect to xx: ddx(x2+y2)=ddx(676) \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(676) 2x+2ydydx=0 2x + 2y \frac{dy}{dx} = 0

STEP 4

Solve for dydx\frac{dy}{dx}.
2x+2ydydx=0 2x + 2y \frac{dy}{dx} = 0 dydx=xy \frac{dy}{dx} = -\frac{x}{y}

STEP 5

Set the slope of the tangent equal to 512\frac{5}{12}.
xy=512 -\frac{x}{y} = \frac{5}{12}

STEP 6

Solve for yy in terms of xx.
xy=512 \frac{x}{y} = -\frac{5}{12} y=125x y = -\frac{12}{5}x

STEP 7

Substitute y=125xy = -\frac{12}{5}x into the circle equation x2+y2=676x^2 + y^2 = 676.
x2+(125x)2=676 x^2 + \left(-\frac{12}{5}x\right)^2 = 676 x2+14425x2=676 x^2 + \frac{144}{25}x^2 = 676

STEP 8

Combine like terms and solve for xx.
x2+14425x2=676 x^2 + \frac{144}{25}x^2 = 676 x2(1+14425)=676 x^2 \left(1 + \frac{144}{25}\right) = 676 x2(25+14425)=676 x^2 \left(\frac{25 + 144}{25}\right) = 676 x2(16925)=676 x^2 \left(\frac{169}{25}\right) = 676 x2=67625169 x^2 = \frac{676 \cdot 25}{169} x2=16900169 x^2 = \frac{16900}{169} x2=100 x^2 = 100 x=±10 x = \pm 10

STEP 9

Find the corresponding yy-values using y=125xy = -\frac{12}{5}x.
For x=10x = 10: y=12510 y = -\frac{12}{5} \cdot 10 y=24 y = -24
For x=10x = -10: y=125(10) y = -\frac{12}{5} \cdot (-10) y=24 y = 24

STEP 10

Identify the points with the smaller and larger yy-values from the solutions.
The points are: - Smaller yy-value: (10,24)(10, -24) - Larger yy-value: (10,24)(-10, 24)
Solution: Smaller yy-value (x,y)=(10,24)(x, y) = (10, -24) Larger yy-value (x,y)=(10,24)(x, y) = (-10, 24)

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