JEE Main 2006 — Coordinate Geometry

A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is

If the lines $3 x-4 y-7=0$ and $2 x-3 y-5=0$ are two diameters of a circle of area $49 \pi$ square units, the equation of the circle is

If $\left(a, a^2\right)$ falls inside the angle made by the lines $y=\frac{x}{2}, x>0$ and $y=3 x, x>0$, then a belongs to

The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z=c^{\prime} y+d^{\prime}$ are perpendicular to each other if

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

The locus of the vertices of the family of parabolas $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ is

Let $C$ be the circle with centre $(0,0)$ and radius 3 units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 \pi}{3}$ at its centre is