A given line has an origin of 5 and forms a 22º angle with the X axis, and is tangent with a point P on a given circle. What are the coordinates of P?

Question

A given line has an origin of 5 and forms a 22º angle with the X axis. It is also tangent with a point P on a given circle. What are the coordinates of P?

Answer 1

HINT

We need to find points on the circle with slope of the tangent equal to the slope of the line.

From the figure we have that the circle is centered in the origin thus to find P it suffice to find the equation of the line through the origin and ortogonal to the given line.

Remember that the condition for orthogonality is

$$m_1\cdot m_2=-1\implies m_2=-\frac{1}{m_1}$$

with

• $m_1=-\arctan22°$ is the slope of the given line
• $m_2$ is the slope of the line orthogonal to the given

Then P can be found by the intersection of the two lines.