Question

the locus of the moving point P such that 2PA=3PB where A is (0,0)& B is (4,-3) is options are a) 5x2+5y2+72x+54y+225=0 b) 5x2+5y2-72x-54y+225=0 please along with solution answer the question

Answer 1

let the point p be (h, k)
Now 2PA,=3PB
PA=√[h^2+k^2]
pb=√[h-4]^2+[k+3]^2
given 2PA=3PB
SQ. both sides
4PAsq. =9PBsq
4(h^2+k^2)=9[(h-4)^2+(k+3)^2]
solve this u will get ans

Answer 2

Converting in to current coordinate the locus of the point P will be  5x² + 5y² - 72x + 54y + 225 = 0.

Step-by-step explanation:

Let the coordinates of P at any instance is (h,k).

Now, the distance between A(0,0) and (P(h,k) is PA = $\sqrt{(h - 0)^{2} + (k - 0)^{2}} = \sqrt{h^{2} + k^{2}}$

And the distance between the points B(4,-3) and P(h,k) is PB = $\sqrt{(h - 4)^{2} + (k + 3))^{2}} = \sqrt{h^{2} + k^{2} - 8h + 6k + 25}$

Now, given that 2(PA) = 3(PB)

Squaring both sides we get 4(PA)² = 9(PB)²

⇒ 4(h² + k²) = 9(h² + k² - 8h + 6k + 25)

⇒ 5h² + 5k² - 72h + 54k + 225 = 0

Therefore, converting in to current coordinate the locus of the point P will be  5x² + 5y² - 72x + 54y + 225 = 0. (Answer)