JEE Main 2004 — Coordinate Geometry

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x=$ 4 , then the equation of the ellipse is

A line makes the same angle $\theta$, with each of the $x$ and $z$ axis. If the angle $\beta$, which it makes with $y$-axis, is such that $\sin ^2 \beta=3 \sin ^2 \theta$, then $\cos ^2 \theta$ equals

The intercept on the line $y=x$ by the circle $x^2+y^2-2 x=0$ is $A B$. Equation of the circle on $\mathrm{AB}$ as a diameter is

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

If the lines $2 x+3 y+1=0$ and $3 x-y-4=0$ lie along diameters of a circle of circumference $10 \pi$, then the equation of the circle is

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $A B C$. If the centroid of this triangle moves on the line $2 x+3 y=1$, then the locus of the vertex $C$ is the line

A variable circle passes through the fixed point $A(p, q)$ and touches $x$-axis. The locus of the other end of the diameter through $\mathrm{A}$ is

If a circle passes through the point $(a, b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is

If one of the lines given by $6 x^2-x y+4 c y^2=0$ is $3 x+4 y=0$, then $c$ equals