Find the Tangent Given the Point (3,4)

Problem

x2+y2=25,(3,4)

Solution

1. Identify the equation of the circle passing through the point (3,4) with the center at the origin (0,0) which is x2+y2=r2

2. Verify the radius by substituting the point into the equation: 3+4=9+16=25 so r2=25

3. Differentiate the equation implicitly with respect to x to find the slope of the tangent line.

d()/d(x)*(x2+y2)=d()/d(x)*25

2*x+2*yd(y)/d(x)=0

4. Solve for the derivative d(y)/d(x) to determine the slope formula.

d(y)/d(x)=−x/y

5. Substitute the coordinates of the point (3,4) into the derivative to find the specific slope m

m=−3/4

6. Apply the point-slope formula y−(y_1)=m*(x−(x_1)) using the point (3,4) and the slope m=−3/4

y−4=−3/4*(x−3)

7. Simplify the equation into slope-intercept form.

y−4=−3/4*x+9/4

y=−3/4*x+9/4+16/4

y=−3/4*x+25/4

Final Answer

y=−3/4*x+25/4

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