A point moves so that square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x -12y = 3. The equation of its locus is .................

Locus of the mid points of the portion of the line \(x\sin\theta+y\cos\theta=p\) intercepted between the axis is ............

Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos \(\alpha\) + y sin \(\alpha\) = p is \(\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}\) where p is a constant.

The line \(\frac{x}{a}+\frac{x}{b}=1\) moves in such a way that \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2},\) where c is a constant. Find the locus of the foot of the perpendicular from the origin on the given line.

If the equation 12x²-10xy+2y²+14x-5y+c=0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.