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A $(-2, 3)$ and B $(4, 1)$ are end points of the diameter of a semi-circle. The semi-circle intersects y-axis at point P. Find the co-ordinates of the point P.
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Solution:
Coordinates of centre O are $(\frac{-2+4}{2}, \frac{3+1}{2})$ i.e.$(1, 2)$ (1/2 Mark)
Let the coordinates of point P be $(0, y)$ (1/2 Mark)
Now $OP^2 = OB^2 \Rightarrow 1^2 + (2 - y)^2 = 3^2 + 1^2$ (1/2 Mark)
$\Rightarrow y = 5$ or $y = -1$ (1/2 Mark)
$\therefore$ Coordinates of P are $(0, 5)$ or $(0, -1)$. (1/2 Mark)
Coordinates of centre O are $(\frac{-2+4}{2}, \frac{3+1}{2})$ i.e.$(1, 2)$ (1/2 Mark)
Let the coordinates of point P be $(0, y)$ (1/2 Mark)
Now $OP^2 = OB^2 \Rightarrow 1^2 + (2 - y)^2 = 3^2 + 1^2$ (1/2 Mark)
$\Rightarrow y = 5$ or $y = -1$ (1/2 Mark)
$\therefore$ Coordinates of P are $(0, 5)$ or $(0, -1)$. (1/2 Mark)