Find a line parallel to a known line that intersects a known circle at one point.

Question

There is a circle with an equation $x^2+y^2=16$ and a line with equation $y=x+1 $. The question is to find an equation of line placed parallel to this line and touching the circle at only one point.

How do I find this out?

Answer 1

For tangency condition the cutting points should coincide,with discriminant of quadratic vanishing.

$ y = x + c, x^2 + y^2 = 16 $ , substitute for y and simplify

$$ 2 x^2 + 2 x c + (c^2 -16) = 0 $$

Discriminant $ \Delta=0, 4 c^2 - 4\; 2\; (c^2 -16) =0 , c = \pm 4 \sqrt{2} $

So there are two tangent straight lines: $ y = x \pm c. $

Answer 2

Draw the line $ y = -x$. This line is perpendicular to the given line and the two tangents.

Substitute $ y = - x $ into the equation of the circle to determine where the line $ y = - x $ intersects the circle. These two points will be where the tangents are. These points are

$$( -2 \sqrt{2} , 2 \sqrt{2} ) \text{ and }( 2 \sqrt{2} , -2 \sqrt{2} )$$

You know that the two equations that intersect those two points have slope = 1. Can you find the equations of both lines?