Tangent of a circle that passes through the origin

RocketMan

New member
Joined
Sep 7, 2014
Messages
1
Hey guys,
I have to find the equations of the lines which pass through the origin and are tangent to the circle (x-2)^2 + (y-1)^2 = 4, and just by drawing it I can tell one of the tangents is x=0. But I can't find the other one. The book says the answer is y=-3/4x and I don't know how to start. Thanks
 
Hey guys,
I have to find the equations of the lines which pass through the origin and are tangent to the circle (x-2)^2 + (y-1)^2 = 4, and just by drawing it I can tell one of the tangents is x=0. But I can't find the other one. The book says the answer is y=-3/4x and I don't know how to start. Thanks
Let the slope of the tangent line = m

Then y = mx is the equation of the tangent line.

Tangent lines will have only one point "in common" (or point of intersection) with the graph.

so the point of intersection is the solution to:

(x-2)2 + (m*x-1)2 = 4

The above is a quadratic equation in "x". Find the value/s of "m" such that the equation has one solution.
 
Hey guys,
I have to find the equations of the lines which pass through the origin and are tangent to the circle (x-2)^2 + (y-1)^2 = 4, and just by drawing it I can tell one of the tangents is x=0. But I can't find the other one. The book says the answer is y=-3/4x and I don't know how to start. Thanks

Using the point slope form for a line
y - y0 = m (x - x0)
where (x0, y0) is a point the line passes through and m is the slope. Thus we have
y = m x
since (y0, y0) =(0, 0)

Since the line we want is tangent to the circle
(x-2)2 + (y-1)2 = 4
and we know, from drawing the problem, that the other solution we want is the one on the lower half of the circle we have:
y = 1 \(\displaystyle -\sqrt{4\space - \space (x-2)^2}\)
From that the slope of the circle can be found and since m will be equal to that slope, you have an equation for m in terms of x.

The value of y (=m x) can also be substituted into the equation for the circle for another value of m in terms of x.

From these values, you can determine a value of x [actually two values, one of which is zero] at which the line is tangent. Use that value of x to determine m.
 
Top