Do I solve this problem alongside the two tangent theorem?

Question

This is the problem: 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.

I am very confused and do not know how to approach this problem let alone solve it.

Answer 1

Let (x,y) be such a point.

The tangent line to the unit circle is perpendicular to the radius from (0,0) to (x,y).

So, (y-2) / (x - 7) = -x / y (Slope of radius is the negative reciprocal of the slope of the line through (x,y) and P = (7,2) ).

Cross multiply to get y² - 2y = -x² + 7x (**)

So, x² + y² = 7x + 2y

Since (x,y) is on the unit circle, x² + y² = 1.

So, 7x + 2y = 1. This tells us that y = (1-7x) / 2.

Substitute into equation ** to get , (1-7x)² / 4 - (1-7x) = -x² + 7x

(1 - 14x + 49x²) / 4 + x² - 1 = 0

(53/4)x² - (14/4)x - 3/4 = 0

53x² - 14x - 3 = 0

x = [14 ± √832] / 106 = 0.404 or -0.140

If x = 0.404, then y = [1 - 7(0.404)] / 2 = -0.914

If x = -0.140, then y = [1 - 7(-0.140)] / 2 = 0.990

The points of tangency are approximately (0.404, -0.914) and (-0.140, 0.990)

Answer 2

Interesting problem. I'm unsure what the questions are that you need to answer but I used trig instead of calculus to find the equations of the lines that are tangent with the unit circle and pass through (7, 2).

Use the fact that the tangent lines intersect the circle radii at right angles. You can calculate the length of the line between the origin and (7, 2) using the distance formula. The length of the radii are 1. This enables you to find the angle between the tangent lines and the line connecting the origin and (7, 2). That gives you the other angles in the triangles. You can use the fact that slope = tangent(angle) and you can find the intersection points of the unit circle and the radii using a system of equations. Once you have the intersection points, you can find the equations of the lines. If you'd like to message me, I'd be happy to show you what I did in detail free of charge.

Calculus (using implicit differentiation) easily gives you the slope of the tangent lines (dy/dx = -x/y) but I didn't quickly see how to use that so I went down the geometry path.