circle, tangent, point problems?????

• Apr 21st 2012, 03:00 PM
sluggerbroth
circle, tangent, point problems?????
How do you solve these problems?
1) find standard equation of circles that pass thru (2,3) and are tangent to both lines 3x-4y=-1 and 4x+3y=7
2 find standard eqations of circles that have centers on 4x+3y=8 and are tangent to both the line x+y=-2 and 7x-y=-6
3) find eqations of lines thru (4,10) and tangent to circle x^2+y^2-4y-36=0

these problems come at the beginning of calculus 1???? help!!!!!!!!
• Apr 21st 2012, 05:00 PM
Prove It
Re: circle, tangent, point problems?????
Quote:

Originally Posted by sluggerbroth
How do you solve these problems?
1) find standard equation of circles that pass thru (2,3) and are tangent to both lines 3x-4y=-1 and 4x+3y=7
2 find standard eqations of circles that have centers on 4x+3y=8 and are tangent to both the line x+y=-2 and 7x-y=-6
3) find eqations of lines thru (4,10) and tangent to circle x^2+y^2-4y-36=0

Hint: Tangents have the same gradient as curves at the point at which they are tangent.
• Apr 21st 2012, 06:10 PM
sluggerbroth
Re: circle, tangent, point problems?????
these problems come at the beginning of calculus 1???? help!!!!!!!!
• Apr 21st 2012, 08:01 PM
Prove It
Re: circle, tangent, point problems?????
Quote:

Originally Posted by sluggerbroth
these problems come at the beginning of calculus 1???? help!!!!!!!!

For the first question, what is the gradient of each of your tangents?
• Apr 22nd 2012, 08:17 AM
emakarov
Re: circle, tangent, point problems?????
One method of solving problem 1 comes from analytic geometry. As you can check, the lines 3x - 4y = -1 and 4x + 3y = 7 are perpendicular and intersect at (1, 1). Let us consider two coordinate systems with the origin (1, 1): one has axes parallel to the original axes (the latter are drawn dashed in the figure below) and the other has the lines 3x - 4y = -1 and 4x + 3y = 7 as axes. The point A(2, 3) in the original (dashed) system has coordinates $x_0 = 1$ and $y_0 = 2$ in the new (x, y)-system.

The circle that touches x'- and y'-axes must have its center O on the angle bisector that has the equation y' = x'. In addition, the distance OC between B and the x'-axis must equal AO. (In this problem, AO happens to be parallel to the x'-axis, but this is a coincidence.) How do we find the coordinates of O? Recall that the locus of points equidistant from the point A and the x'-axis is a parabola. It is easier to find its equation in the system (x'', y') where the y'-axis is shifted so that it goes through A (dashed line). Then $x' = x'' + x_0'$. If O has coordinates (x'', y'), then $AO^2 = x''^2+(y'-y_0')^2$ and $OC=y'^2$. So, $x''^2+(y'-y_0')^2=y'^2$ is the equation of the parabola. The equation of the bisector in the (x'', y')-system is y' = x'' + 2. Solving these two equations allows finding x'' (and hence x') and y' knowing $x_0'$ and $y_0'$.

We know the coordinates $(x_0=1, y_0=2)$ of A in the (x, y)-system. How do we find the coordinates $(x_0',y_0')$ of the same point in the (x', y')-system? Suppose that $(x_1,y_1)$ and $(x_2,y_2)$ are unit vectors along the x'-, y'-axes expressed in terms of x and y. Then

$\binom{x_0}{y_0}=x_0'\binom{x_1}{y_1}+y_0'\binom{x _2}{y_2}$

i.e.,

$x_0=x_1x_0'+x_2y_0'$
$y_0=y_1x_0'+y_2y_0'$ (*)

This allows finding $(x_0,y_0)$ from $(x_0',y_0')$, which is also needed below. To find $(x_0',y_0')$ from $(x_0,y_0)$ there are several ways. First, one can solve the system (*) for $x_0',y_0'$. Second, one can use the formula for the distance of a point $(x_0,y_0)$ from a line $ax+by+c$. This distance is $\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$.

So, here are the instructions.

Step 1. Find the unit vectors $(x_1,y_1)$ and $(x_2,y_2)$ along the x'- and y'-axes.

Step 2. Find the coordinates $(x_0',y_0')$ of A in the (x', y')-system.

Step 3. Find the equation of the parabola in the (x'', y')-system.

Step 4. Find the coordinates of the circle center O in the (x'', y')-system.

Step 5. Convert the coordinates of O into the (x', y')-system and then (x, y)-system using (*).

Step 6. Find the radius OA = OC of the circle and write the circle equation in the (x, y)-system.