Geometry Geometry Math Forum

 November 11th, 2016, 06:39 PM #1 Senior Member   Joined: Nov 2010 From: Indonesia Posts: 1,165 Thanks: 111 [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.
 November 12th, 2016, 01:43 AM #2 Global Moderator   Joined: Dec 2006 Posts: 16,766 Thanks: 1231 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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