Help quick: circle O is determined by the equation x^2 + y^2 = 3. Point P has coordinates (0, -3) Let A = (a, b) be point on circle so...

[kqtzs]

circle O is determined by the equation x^2 + y^2 = 3.
Point P has coordinates (0, -3)

Let A with coordinates (a, b), be a point on circle O so that AP is tangent to the circle.

• Find coordinates (a, b) for such a point A.
• Remember to notate your work and indicate how you are applying any relevant definitions or theorems.

 

[Otis]

Did you start by finding the two x-intercepts and two y-intercepts, in order to sketch the circle and plot point P?

Once you visualize the positions of P and A, think about how to form a right triangle.

 

[Otis]

Let the circle's center be point O. The right triangle that I have in mind has OA as its hypotenuse, and the lengths of its other two sides are a and b. Hint: a^2 and b^2 are Integers.

But maybe you're expected to use a different method. I'm not sure what you've been taught. Instead, we could use the distance formula and then solve a system of two equations (using the substitution method from algebra).

Can you tell us what your class has been doing?

 

[The Highlander]

circle O is determined by the equation x^2 + y^2 = 3.
Point P has coordinates (0, -3)

Let A with coordinates (a, b), be a point on circle O so that AP is tangent to the circle.

• Find coordinates (a, b) for such a point A.
• Remember to notate your work and indicate how you are applying any relevant definitions or theorems.

Here is a diagram that will enable you to "visualize" (as @Otis suggested) where the Point(s) A may lie.

​

You can access the graph in Desmos by clicking here (but, naturally, I've had to remove all the straight lines & Point A as they would just give you the answer. ). So all you will get if you visit Desmos is this...

​

But even that's a "start", innit?

Hope that helps.

 

[The Highlander]

@kqtzs

Let the circle's center be point O. The right triangle that I have in mind has OA as its hypotenuse, and the lengths of its other two sides are a and b. Hint: a^2 and b^2 are Integers.

But maybe you're expected to use a different method. I'm not sure what you've been taught. Instead, we could use the distance formula and then solve a system of two equations (using the substitution method from algebra).

I just used Pythagoras' Theorem (taking OP as my hypotenuse) to find the coordinates of A, so that's another method open to use.

(I expect you must be familiar with that theorem?).

 

[Steven G]

(I expect you must be familiar with that theorem?).

I expect you must be familiar with that theorem?
That's a dangerous assumption to make in my opinion!

 

[The Highlander]

I expect you must be familiar with that theorem?
That's a dangerous assumption to make in my opinion!

You are quite right that there is inherent danger in any "assumption".

However, there was no "assumption" made in the comment you quoted from my post!

I simply said that there was an expectation that the student would already have been taught about Pythagoras' Theorem, especially given the level of difficulty of the original question.

Expecting something to be true is quite different from assuming that it is true.

I did not assume that s/he could (or should) use Pythagoras to solve this problem. I simply observed that it was another method that might be used to solve it (and expected that s/he would know how to use Pythagoras; though s/he might not immediately 'see' how to use it to solve this problem which is why I produced the graph with some (limited) hints in it, eg: omitting to identify a significant right angle that comes from 'Circle Theorems'.).

But thank you for taking the time to read what I wrote. It often seems to me that many people just glance at my posts (and occasionally those of others too), focus on a single element that 'sticks out' to them, and then make unwarranted comments without bothering to read the whole of the post or give due credit to the work that has been done and the resulting help being offered to the OP.

As an 'Elite' member of the forum, I'm sure I can trust that you wouldn't be guilty of that kind of behaviour.

 

[jonah2.0]

Beer induced reaction follows.

circle O is determined by the equation x^2 + y^2 = 3.
Point P has coordinates (0, -3)

Let A with coordinates (a, b), be a point on circle O so that AP is tangent to the circle.

• Find coordinates (a, b) for such a point A.
• Remember to notate your work and indicate how you are applying any relevant definitions or theorems.

For an algebraic perspective, see revived thread at

Circle Tangent question

Any thoughts on this question? The standard way of doing it would be using the discriminant. So the equation of the line is y=m(x-5) and substituting this into the circle equation: This needs tidying up: And then using the discriminant, but the algebra gets horrendous and its meant to be an...

www.freemathhelp.com

Edit: My bad. This thread's requirement is merely the coordinates of point A whereas the other thread's requirements are the tangent lines.