November 11th, 2016, 05:39 PM  #1 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 1,314 Thanks: 114  [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 Xaxis 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 $\displaystyle (xa)^2+(yb)^2=r^2$ and got $\displaystyle 3612a+a^2+b^2=36$. 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, 12:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
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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