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Finding Tangent Lines Through a Point to a Circle

This problem involves finding the equations of the lines that pass through the point (1, 3) and are tangent to the circle defined by the equation x² + y² = 2. The circle has a center at (0, 0) and a radius of √2. We will solve this problem both geometrically and algebraically, showcasing how to derive the tangent lines using the midpoint between the circle's center and the point, and finding the points of intersection of the relevant circles. The resulting tangent line equations will be identified and explained.

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Finding Tangent Lines Through a Point to a Circle

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  1. Problem Set 2, Problem # 2 Ellen Dickerson

  2. Problem Set 2, Problem #2 Find the equations of the lines that pass through the point (1,3) and are tangent to the circle with equation x2 + y2 = 2

  3. A Circle with equation x2 + y2 = 2 has center point (0,0) and radius √(2) draw/show point (1,3) and how there are two lines that go through (1,3) and are tangent to the circle.

  4. Solving the Problem Geometrically (draw picture)

  5. Solving the Problem Algebraically • Midpoint (between the center of the original circle and the point (1,3)) • ((1+0)/2 , (3+0)/2) = ((1/2), (3/2)) • Distance (between the center of the original circle and the midpoint) • (√(12+32))/2 = (√10)/2 • Formula for the new circle • (x-(1/2)2 +(y-(3/2)2 = 5/2

  6. Finding the points where the circles intersect (x-(1/2)2 +(y-(3/2)2 = 5/2 x2+y2 = 2 x2–x+y2-3y = 0 + -1(x2+y2)=-1 (2) -x-3y = -2 x = 2-3y Plug 2-3y into x in the equationx2+y2 = 2 (2-3y)2+y2 = 2 10y2-12y+2 = 0 (5y-1)(2y-2)= 0 5y-1= 0 and 2y-2= 0 5y=1 2y=2 y=(1/5) and y=1

  7. Finding X (use formula x = 2-3y) y=(1/5) x=2-3(1/5) x=2-(3/5) x=(7/5) = 1.4 ( (7/5), (1/5 )) y=1 x=2-3(1) x=2-3 x=-1 (-1,1)

  8. Finding the line ((7/5), (1/5 )) and (1,3) m=(1/5)-3/(7/5)-1 m=-(14/5)/(2/5) m=-7 y-3=-7(x-1) y-3 =-7x+(7) y= -7x +10 (-1,1), and (1,3) m=(1-3)/(-1-1) m=1 y-3=(1)(x-3) y-3=(x-1) y= x+2

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