Tangent line to a circle
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Have you worked an exercise like this using Real numbers for the coordinates at (x', y') and for the radius a ?
The overall approach is the same, when using symbols for these three numbers. (I'm wondering whether or not the abstract symbolism is the roadblock, for you.)
If you have not done an exercise like this before, then you need to know the following.
The radius is perpendicular to the tangent line at any point on a circle. It's easy to find the slope of the radius. And, you should already know that slopes of perpendicular lines are negative reciprocals of each other, yes? That gives us the slope of the tangent line.
Since you did not ask any questions, I can't really know why you're stuck. So, I'll give you the steps, and maybe you'll give us some specific information about your situation.
(1) Calculate the slope of the radial line segment from (0, 0) to (x', y')
(2) Take the negative reciprocal, to determine the slope of the tangent line
(3) Use the Point-Slope formula with the given point (x', y') and the slope expression from step (2)
(4) Multiply both sides by y'
(5) Distribute the factor of -x' on the RHS
(6) Add (y')^2 to both sides
(7) On the RHS, substitute a^2 for (x')^2 + (y')^2
(8) Add x'x to both sides
If you get stuck (even if it's at step one), please show what you tried, or explain what you don't understand, and somebody will clarify.
Cheers ~ Mark
The overall approach is the same, when using symbols for these three numbers. (I'm wondering whether or not the abstract symbolism is the roadblock, for you.)
If you have not done an exercise like this before, then you need to know the following.
The radius is perpendicular to the tangent line at any point on a circle. It's easy to find the slope of the radius. And, you should already know that slopes of perpendicular lines are negative reciprocals of each other, yes? That gives us the slope of the tangent line.
Since you did not ask any questions, I can't really know why you're stuck. So, I'll give you the steps, and maybe you'll give us some specific information about your situation.
(1) Calculate the slope of the radial line segment from (0, 0) to (x', y')
(2) Take the negative reciprocal, to determine the slope of the tangent line
(3) Use the Point-Slope formula with the given point (x', y') and the slope expression from step (2)
(4) Multiply both sides by y'
(5) Distribute the factor of -x' on the RHS
(6) Add (y')^2 to both sides
(7) On the RHS, substitute a^2 for (x')^2 + (y')^2
(8) Add x'x to both sides
If you get stuck (even if it's at step one), please show what you tried, or explain what you don't understand, and somebody will clarify.
Cheers ~ Mark