JEE Main 2013 — Coordinate Geometry

If $a$ and $c$ are positive real numbers and the ellipse $\frac{x^2}{4 c^2}+\frac{y^2}{c^2}=1$ has four distinct points ir common with the circle $x^2+y^2=9 a^2$, then

If each of the lines $5 x+8 y=13$ and $4 x-y=3$ contains a diameter of the circle $x^2+y^2-2\left(a^2-7 a+11\right)$ $x-2\left(a^2-6 a+6\right) y+b^3+1=0$, then :

If the $x$-intercept of some line $L$ is double as that of the line, $3 x+4 y=12$ and the $y$-intercept of $L$ is half as that of the same line, then the slope of $L$ is :

If the circle $x^2+y^2-6 x-8 y+\left(25-a^2\right)=0$ touches the axis of $x$, then a equals.

If the image of point $\mathrm{P}(2,3)$ in a line $\mathrm{L}$ is $\mathrm{Q}(4,5)$, then the image of point $\mathrm{R}(0,0)$ in the same line is:

If the three lines $x-3 y=p, a x+2 y=q$ and $a x+y=r$ form a right-angled triangle then :

If two lines $L_1$ and $L_2$ in space, are defined by $$ \begin{aligned} & L_1=\{x=\sqrt{\lambda} y+(\sqrt{\lambda}-1), \\ & \quad z=(\sqrt{\lambda}-1) y+\sqrt{\lambda}\} \text { and } \\ & L_2=\{x=\sqrt{\mu} y+(1-\sqrt{\mu}), \end{aligned} $$ $$ z=(1-\sqrt{\mu}) y+\sqrt{\mu}\} $$ then $L_1$ is perpendicular to $L_2$, for all nonnegative reals $\lambda$ and $\mu$, such that :

Let $\theta_1$ be the angle between two lines $2 x+3 y+$ $c_1=0$ and $-x+5 y+c_2=0$ and $\theta_2$ be the angle between two lines $2 x+3 y+c_1=0$ and $-x+5 y+$ $c_3=0$, where $c_1, c_2, c_3$ are any real numbers : Statement-1: If $c_2$ and $c_3$ are proportional, then $\theta_1=\theta_2$. Statement-2: $\theta_1=\theta_2$ for all $c_2$ and $c_3$.

Let the equations of two ellipses be $$ E_1: \frac{x^2}{3}+\frac{y^2}{2}=1 \text { and } E_2: \frac{x^2}{16}+\frac{y^2}{b^2}=1 \text {, } $$ If the product of their eccentricities is $\frac{1}{2}$, then the length of the minor axis of ellipse $E_2$ is :

Let $x \in(0,1)$. The set of all $x$ such that $\sin ^{-1} x>\cos ^{-1} x$, is the interval:

Statement-1: The line $x-2 y=2$ meets the parabola, $y^2+2 x=0$ only at the point $(-2,-2)$. Statement-2: The line $y=m x-\frac{1}{2 m}(m \neq 0)$ is tangent to the parabola, $y^2=-2 x$ at the point $\left(-\frac{1}{2 m^2},-\frac{1}{m}\right)$

Statement 1: The only circle having radius $\sqrt{10}$ and a diameter along line $2 x+y=5$ is $x^2+y^2-6 x$ $+2 y=0$. Statement $2: 2 x+y=5$ is a normal to the circle $x^2+y^2-6 x+2 y=0$.

Statement-1: The slope of the tangent at any point $\mathrm{P}$ on a parabola, whose axis is the axis of $x$ and vertex is at the origin, is inversely proportional to the ordinate of the point $\mathrm{P}$. Statement-2: The system of parabolas $y^2=4 a x$ satisfies a differential equation of degree 1 and order 1.

The acute angle between two lines such that the direction cosines $l, m, n$, of each of them satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is :

The circle passing through $(1,-2)$ and touching the axis of $x$ at $(3,0)$ also passes through the point

The $x-$coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as$(0,1),(1,1)$ and $(1,0)$ is

The equation of the circle passing through the foci of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}=1$, and having centre at $(0,3)$ is