Coordinate Geometry Questions

If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$:

If the equation of the locus of a point equidistant from the points $(a_1,b_1)$ and $(a_2,b_2)$ is $(a_1-a_2)x+(b_1-b_2)y+c=0$, then the value of '$c$ ' is:

Locus of centroid of the triangle whose vertices are $(a\cos t,a\sin t)$, $(b\sin t,-b\cos t)$ and $(1,0)$, where $t$ is a parameter, is:

If the pair of straight lines $x^2-2pxy-y^2=0$ and $x^2-2qxy-y^2=0$ be such that each pair bisects the angle between the other pair, then:

A square of side $a$ lies above the $x$ -axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha$ ($0<\alpha <\pi /4$) with the positive direction of $x$ -axis. The equation of its diagonal not passing through the origin is:

If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points, then:

The lines $2x-3y=5$ and $3x-4y=7$ are diameters of a circle having area as 154 sq units. Then the equation of the circle is:

The normal at the point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$, then: