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November 11th, 2016, 05:39 PM   #1
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[ASK] About Circles in Coordinates

3. A(a,b), B(-a,-b), and C is plane XOY. P moves along with curve C. If the multiplication product of PA's and PB's gradients are always k, C is a circle only if k = ...?
4. The radius of a circle which meets X-axis at (6,0) and meets the curve $\displaystyle y=\sqrt{3x}$ at one point is ....
5. A circle meets the line x + y = 3 at (2,1). It also meets the point (6,3). Its radius is ....

I have no idea how to do number 3, or even the meaning.

For number 4, I substituted x = 6 and y = 0 to the equation $\displaystyle (x-a)^2+(y-b)^2=r^2$ and got $\displaystyle 36-12a+a^2+b^2=36$.

For number 5, I substituted both coordinates to the circle equation and got 2a + b = 10.

Please help how to continue each number.
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November 12th, 2016, 12:43 AM   #2
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3. If P is the point (x, y), PA and PB have gradients (y - b)/(x - a) and (y + b)/(x + a) respectively.

If the product of those gradients is k, (y² - b²)/(x² - a²) = k, and so -kx² + y² = -ka² + b².

Can you finish from there?

Note that PA and PB needn't both have gradients.
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