JEE Main 2020 — Coordinate Geometry

For $a>0,$ let the curves ${C}_{1}:{y}^{2}=ax$ and ${C}_{2}:{x}^{2}=ay$ intersect at origin $O$ and a point $P.$ Let the line $x=b(0<b<a)$ intersect the chord $OP$ and the $x$ -axis at points $Q$ and $R,$ respectively. If the line $x=b$ bisects the area bounded by the curves, ${C}_{1}$ and ${C}_{2},$ and the area of $\Delta OQR=\frac{1}{2},$ then ‘ $a$ ’ satisfies the equation:

A ray of light coming from the point $(2,2\sqrt{3})$ is incident at an angle $30^{\circ}$ on the line $x=1$ at the point $A$. The ray gets reflected on the line $x=1$ and meets $x$ -axis at the point $B$. Then, the line $AB$ passes through the point

If a line $y=mx+c$, is a tangent to the circle ${(x-3)}^{2}+{y}^{2}=1$, and it is perpendicular to a line ${L}_{1},$ where ${L}_{1}$ is the tangent to the circle ${x}^{2}+{y}^{2}=1$, at the point $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$, then

If one end of a focal chord $AB$ of the parabola ${y}^{2}=8x$ is at $A(\frac{1}{2},-2),$ then the equation of the tangent to it at $B$ is:

If the length of the chord of the circle, ${x}^{2}+{y}^{2}={r}^{2}(r>0)$ along the line, $y-2x=3$ is $r$, then ${r}^{2}$ is equal to:

If a $\Delta ABC$ has vertices $A(–1,7),B(–7,1)$ and $C(5,–5)$, then its orthocentre has coordinates:

For some $\theta \in (0,\frac{\pi }{2}),$ if the eccentricity of the hyperbola, ${x}^{2}-{y}^{2}{\mathrm{sec}}^{2}\theta =10$ is $\sqrt{5}$ times the eccentricity of the ellipse, ${x}^{2}{\mathrm{sec}}^{2}\theta +{y}^{2}=5,$ then the length of the latus rectum of the ellipse, is

Let the tangents drawn from the origin to the circle, ${x}^{2}+{y}^{2}-8x-4y+16=0$ touch it at the points $A$ and $B$ . Then ${(AB)}^{2}$ is equal to

A triangle $ABC$ lying in the first quadrant has two vertices as $A(1,2)$ and $B(3,1)$. If$\angle BAC={90}^{o},$and $ar(\Delta \mathrm{ABC})=5\sqrt{5}$ sq. units, then the abscissa of the vertex $C$ is :

The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points$(1,2)$ and $(\mathrm{sin}\theta ,\mathrm{cos}\theta )$ lie on the same side of the line $x+y=1$ is?

If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k,3)$ has $y$-intercept equal to $-4$, then a value of $k$ is;

Let two points be $A(1,-1)$ and $B(0,2).$ If a point $P(x\text{'},y\text{'})$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line $3x+y-4\lambda =0,$ then a value of $\lambda$ is

Let the latus rectum of the parabola ${y}^{2}=4x$ be the common chord to the circles ${C}_{1}$ and ${C}_{2}$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles ${C}_{1}$ and ${C}_{2}$ is :

The numbers of integral values of $k$ for which the line, $3x+4y=k$ intersects the circle, ${x}^{2}+{y}^{2}-2x-4y+4=0$ at two distinct points is....

The area (in sq. units) of an equilateral triangle inscribed in the parabola ${y}^{2}=8x,$ with one of its vertices on the vertex of this parabola is

Let $A(1,0),B(6,2)$ and $C(\frac{3}{2},6)$ be the vertices of a triangle $ABC.$ If $P$ is a point inside the triangle $ABC$ such that the triangles $APC,APB$ and $BPC$ have equal areas, then the length of the line segment $PQ,$ where $Q$ is the point $(-\frac{7}{6},-\frac{1}{3}),$ is

Let $L$ denote the line in the $xy$-plane with $x$ and $y$ intercepts as $3$ and $1$ respectively. Then the image of the point$(-1,-4)$ in the line is :

Let $PQ$ be a diameter of the circle ${x}^{2}+{y}^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is _______

If the line, $2x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4x-2y+\alpha =0$ and $6x-3y+\beta =0$ respectively, then the sum of all possible values of $\alpha$ and $\beta$ is ____________.

If the distance between the foci of an ellipse is $6$ and the distance between its directrix is $12,$ then the length of its latus rectum is

A hyperbola having the transverse axis of length, $\sqrt{2}$ has the same foci as that of the ellipse, $3{x}^{2}+4{y}^{2}=12$ then this hyperbola does not pass through which of the following points?

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola ${x}^{2}=4y$ internally in the ratio $1:2$ is:

If ${e}_{1}$ and ${e}_{2}$ are the eccentricities of the ellipse $\frac{{x}^{2}}{18}+\frac{{y}^{2}}{4}=1$ and the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$ respectively and $({e}_{1},{e}_{2})$ is a point on the ellipse $15{x}^{2}+3{y}^{2}=k$ , then the value of $k$ is equal to

The locus of the centre of a circle which touches the line x=1 and passes through the origin is:

Let ${e}_{1}$ and ${e}_{2}$ be the eccentricities of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{{b}^{2}}=1(b<5)$ and the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{{b}^{2}}=1$ respectively satisfying ${e}_{1}{e}_{2}=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to:

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$, to the line $x=y$, is.

If the co-ordinates of two points $A$ and $B$ are $(\sqrt{7},0)$ and $(-\sqrt{7},0)$ respectively and $P$ is any point on the conic, $9{x}^{2}+16{y}^{2}=144,$ then $PA+PB$ is equal to :

The diameter of the circle, whose Centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$ is