Find the equations and points for tangent lines touching a unit circle and with a point P=(7,2).

Question

Draw the unit circle and plot the point P=(7,2). Observe there are TWO lines tangent to the circle passing through the point P. Answer the questions below with 3 decimal places of accuracy.

(a) The line L1 is tangent to the unit circle at the point:

(b) The tangent line L1 has equation:

(c) The line L2 is tangent to the unit circle at the point:

(d) The tangent line L2 has equation:

 

I have tried this problem about four times and i just cannot figure it out, can someone please help me.

 

Answer 1

Draw and label of diagram.

O is the center of the circle

T₁ is the tangent point at the top

T₂ is the tangent point at the bottom

 

By Pythagoras

PO = √53

T₁P = T₂P = 2√13

 

T₁ and T₂ are the points of intersection (common solutions ) of two circles:

x² + y² = 1   (the unit circle) and

52 = (x - 7)² + (y - 2)²

 

Determine the the common solutions.