Question

A point on the y-axis, whose distances from the points A(6,7) and B(4, -3) are in the ratio 1 : 2 is__________.

• A. (9, 0) or (0,$\frac{35}{3}$)
• B. (0, 9) or (0,$\frac{-35}{3}$)
• C. (0, 9) or (0,$\frac{35}{3}$)
• D. (0, -9) or (0,$\frac{35}{3}$)

Answer 1

The correct option is C

(0, 9) or (0,$\frac{35}{3}$)

Let the point P(0, y) be the point on y - axis whose distance from the point A(6, 7) and B(4, -3) are in the ratio 1 : 2

$\therefore \frac{PA}{PB}=\frac{1}{2}$

$\frac{(\sqrt{(6{)}^2+(y-7{)}^2}}{(\sqrt{(4{)}^2+(y-(-3){)}^2}}$ = $\frac{1}{2}$

Squaring both sides we get,

$\frac{36+y^2-14y+9}{16+y^2+6y}+9=\frac{1}{4}$

$4(y^2-14y+85)=y^2+6y+25$

$4y^2-56y+340=y^2+6y+25$

$3y^2-62y+315=0$

$3y^2-27y-35y+315=0$

$3y(y-9)-35(y-9)=0$

$(y-9)(3y-35)=0$

$y-9=0or3y-35=0$

$y=9ory=\frac{35}{3}$

Therefore the point on the y-axis is (0,9) or (0,$\frac{35}{3}$)