Problem Set 2, Problem # 2 Ellen Dickerson

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 x 2 + y 2 = 2

A Circle with equation x 2 + y 2 = 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.

Solving the Problem Geometrically (draw picture)

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) (√( ))/2 = (√10)/2 Formula for the new circle (x-(1/2) 2 +(y-(3/2) 2 = 5/2

Finding the points where the circles intersect (x-(1/2) 2 +(y-(3/2) 2 = 5/2 x 2 +y 2 = 2 x 2 –x+y 2 -3y = (x 2 +y 2 )=-1 (2) -x-3y = -2 x = 2-3y Plug 2-3y into x in the equation x 2 +y 2 = 2 (2-3y) 2 +y 2 = 2 10y 2 -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

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)

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