Quick Locus Question? (1 Viewer)

[Zokunu]

littlehelp?

(a) Given the points (4,0), B (-2,0) and P (x,y), find the gradients of AP and BP.

AP = y/x-4
BP = y/x+2

(b) Hence show that the locus of the point P(x,y) that moves so the angle APB is a right angle has equation x^2 + y^2 - 2x - 9 = 0.

(c) Complete the square in x, and hence describe the locus geometrically.

Stepbystep instructions would be nice ,thanks

 

[Carrotsticks]

For (b), multiply the gradients and let your expression be equal to -1. This is because we have a right angle, so m_1 x m_2 = -1.

For (c),

 

[Zokunu]

For (b), multiply the gradients and let your expression be equal to -1. This is because we have a right angle, so m_1 x m_2 = -1.

For (c),

Hey man, thanks but i think (c) is wrong. I check and it's meant to be (x-1)^2 + y^2 = 9. . But thanks anyways . can u help me out with one more question?

Find the locus of the point P (x,y) whose distance from the point A (4,0) is always twice it's distance from the point B (1,0)

PA^2 = 4 x PB^2
(x-4)^2 + y^2 = 4(x-1)^2 + 4y^2

x^2 - 8x + 8 + y^2 = 4x^2 - 8x + 4 + 4y^2
o=3x^2 - 4 + 3y^2
3x^2 + 3y^2 = 4.

Answer: x^2 + y^2 = 4.

What am i doing wrong here....?

 

[Carrotsticks]

Hey man, thanks but i think (c) is wrong. I check and it's meant to be (x-1)^2 + y^2 = 9. . But thanks anyways . can u help me out with one more question?

Find the locus of the point P (x,y) whose distance from the point A (4,0) is always twice it's distance from the point B (1,0)

PA^2 = 4 x PB^2
(x-4)^2 + y^2 = 4(x-1)^2 + 4y^2

x^2 - 8x + 8 + y^2 = 4x^2 - 8x + 4 + 4y^2
o=3x^2 - 4 + 3y^2
3x^2 + 3y^2 = 4.

Answer: x^2 + y^2 = 4.

What am i doing wrong here....?

According to the expression you provided, which was x^2 + y^2 - 2x - 9 = 0, the equivalent expression (by completing the square) is indeed (x-1)^2 + y^2 = 10. The only way to get the expression you provided is if we have x^2 + y^2 - 2x - 8 = 0.

For the above problem, I have bolded where you made the error.

 

[Zokunu]

According to the expression you provided, which was x^2 + y^2 - 2x - 9 = 0, the equivalent expression (by completing the square) is indeed (x-1)^2 + y^2 = 10. The only way to get the expression you provided is if we have x^2 + y^2 - 2x - 8 = 0.

For the above problem, I have bolded where you made the error.

oh soz... ur right. I made a typo. It is x^2 + y^2 - 2x - 8 = 0. My bad.

And thanks I got it now