Answer to Question #91460 in Analytic Geometry for Charity

Question

determine the coordinates of P and Q if the sketch shows the hyperbola defined by y=4/9x;the straight line defined by y=x;a circle with a centre at P,touching the x-axis and y-axis at R and S ,respectively ;and the straight line through the point T,S and R. The line joining T and Q is parallel to the y-axis

Answer 1

Let

$$P(x_P, y_P), Q=(x_Q, y_Q)$$

Since the hyperbola is defined by y=4/9x and the straight line is defined by y=x, then

$$y_P={4 \over 9x_P}, y_P=x_P, x_P>0, y_P>0$$

$$y_Q={4 \over 9x_Q}, y_Q=x_Q, x_Q<0, y_Q<0$$

$$x_P=y_P=\sqrt{4 \over 9}={2 \over 3}$$

$$x_Q=y_Q=-\sqrt{4 \over 9}=-{2 \over 3}$$

$$P({2 \over 3}, {2 \over 3}), Q=(-{2 \over 3}, -{2 \over 3})$$