Let $A(5,-3), B(3,-2), C(-1,5)$ be three points. If $P$ is a point satisfying the condition $P A^2+2 P B^2=3 P C^2$, then a point that lies on the locus of $P$ is

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$, if the equation $49 x^2+25 y^2=1225$ is transformed to $p x^2+q x y+r y^2=t$ and the GCD of $p, q, r, t$ is 1 , then

Let the slope of a diameter $A C$ of a circle of radius 25 units be $\frac{3}{4}$. If $(3,2)$ is the centre of the circle, $A=\left(x_1, y_1\right)$ and $C=\left(x_2, y_2\right)$, then $\frac{x_1 x_2}{y_1 y_2}=$

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1$, then $\sin \theta=$