Find center of circle given circle radius, tangent line and a point lies on the circle.

Question

Question

Find center of circle given circle radius, tangent line and a point lies on the circle

Based on the image, is it possible to find the center point of circle $(h,k)$, given the radius of circle is $\sqrt{17}$, the tangent line is $y=-(\frac{-1}4x)+(\frac 92)$ and a point lies on the circle $(1,2)$.

Answer 1

Let us look at the geometry behind the problem.

1. The center must lie on a line parallel to the tangent at a perpendicular distance equal to the radius $\sqrt{17}$. It also lies in the same half-plane as the given point $(1,2)$, so the parallel line is to be drawn below and to the left of the given tangent. Note carefully that the distance $\sqrt{17}$ is the perpendicular distance, so the vertical or horizontal displacement you would use to give an equation for the parallel line would be greater. Can you figure out how much greater given the slope $-1/4$? You need to do that to get an equation for the parallel line in terms of $x$ and $y$.

2. The center is also $\sqrt{17}$ units from the given point $(1,2)$ and so lies on the circle having the radius and center indicated by this fact. You should be able to get the equation for this auxiliary circle from this fact.

3. Your auxiliary circle and parallel line will then intersect in two points, either of which would then qualify as the center of the circle you want. From the drawing it looks like you want the point towards the left, with a negative $x$- coordinate for the center.