JEE Main 2019 — Coordinate Geometry

A point $P$ moves on the line $2x-3y+4=0.$ If $Q(1, 4)$ and $R(3, -2)$ are fixed points, then the locus of the centroid of $\Delta PQR$ is a line:

A point on the straight line, $3x+5y=15$ which is equidistant from the coordinate axes will lie only in:

A rectangle is inscribed in a circle with a diameter lying along the line $3y=x+7.$ If the two adjacent vertices of the rectangle are $(-8,5)$ and $(6,5),$ then the area of the rectangle $($in sq. units$)$ is:

A square is inscribed in the circle $x^{2}+y^{2}-6 x+8 y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

A straight line $L$ at a distance of $4$ units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^{\circ}$ with the line $x+y=0.$ Then an equation of the line $L$ is: Note: In actual JEE Main paper, two options were correct for this question. Hence, we have changed one option.

A triangle has a vertex at $(1, 2)$ and the mid points of the two sides through it are $(-1, 1)$ and $(2, 3)$ . Then the centroid of this triangle is:

An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length $4$ , passes through which of the following points?

Axis of a parabola lies along $x$-axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin, on the positive $x$-axis then which of following points does not lie on it?

Consider the set of all lines $px+qy+r=0$ such that $3p+2q+4r=0.$ Which one of the following statements is true?

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

If a directrix of a hyperbola centered at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then:

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 , then the eccentricity of the hyperbola is:

If a straight line passing through the point $P(-3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is :

If a tangent to the circle ${x}^{2}+{y}^{2}=1$ intersects the coordinate axes at distinct points $P$ and $Q,$ then the locus of the mid-point of $PQ$ is:

If a variable line $3x+4y-\lambda =0$ is such that the two circles ${x}^{2}+{y}^{2}-2x-2y+1=0$ and ${x}^{2}+{y}^{2}-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval :

If in a parallelogram $\mathrm{ABDC}$, the coordinates of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are respectively (1,2),(3,4) and $(2,5),$ then the equation of the diagonal $\mathrm{AD}$ is :

If $5x+9=0$ is the directrix of the hyperbola $16{x}^{2}-9{y}^{2}=144,$ then its corresponding focus is:

If one end of a focal chord of the parabola, ${y}^{2}=16x$ is at $(1,4)$, then the length of this focal chord is

If the angle of intersection at a point where the two circles with radii $5 cm$ and $12 cm$ intersect is $90^{\circ}$, then the length (in cm) of their common chord is:

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^{2}+4\left(x-a^{2}\right)=0$ and the other two vertices are the points of intersection of the parabola and $y$ -axis, is 250 sq. units, then a value of 'a' is :

If the circles ${x}^{2}+{y}^{2}+5Kx+2y+K=0$ and $2({x}^{2}+{y}^{2})+2Kx+3y-1=0,(K\in R)$, intersect at the points P and Q, then the line $4x+5y-K=0$, passes through $P$ and $Q,$ for:

If the circles ${x}^{2}+{y}^{2}-16x-20y+164={r}^{2}$ and ${(x-4)}^{2}+{(y-7)}^{2}=36$ intersect at two distinct points, then:

If the line $3x+4y-24=0$ intersects the $x$-axis is at the point $A$ and the $y$-axis at the point $B,$ then the incentre of the triangle $OAB,$ where $O$is the origin, is:

If the straight line $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta )$, then $\beta$ equals :

If the two lines $x+(a-1)y=1$ and $2x+{a}^{2}y=1,(a\in R-{0,1})$ are perpendicular, then the distance of their point of intersection from the origin is

If the vertices of a hyperbola be at $(-2, 0)$ and $(2, 0)$ and one of its foci be at $(-3, 0)$, then which one of the following points does not lie on this hyperbola ?

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5\sqrt{3}),$ then the length of its latus rectum is:

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, ${y}^{2}=4x.$ Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\Delta ACB$ is maximum. Then, the area (in sq. units) of $\Delta ACB$ , is:

Let ${C}_{1}$ and ${C}_{2}$ be the centres of the circles ${x}^{2}+{y}^{2}-2x-2y-2=0$ and ${x}^{2}+{y}^{2}-6x-6y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P{C}_{1} Q{C}_{2}$ is :

Let $S$ and ${S}^{'}$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta {S}^{'}BS$ is a right angled triangle with right angle at $B$ and area $(\Delta {S}^{'}BS)=8\mathrm{sq}.\mathrm{units}$, then the length of a latus rectum of the ellipse is :

Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then, the locus of a point $P$ such that the perimeter of $\Delta AOP$ is $4$ is

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, ${y}^{2}=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $\Delta PXQ$ is maximum. Then this maximum area (in sq. units) is :

Let $S$ be the set of all triangles in the $xy$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is:

Let $0<\theta <\frac{\pi }{2}.$ If the eccentricity of the hyperbola $\frac{{x}^{2}}{{\mathrm{cos}}^{2}⁡\theta }-\frac{{y}^{2}}{{\mathrm{sin}}^{2}⁡\theta }=1$ is greater than $2,$ then the length of its latus rectum lies in the interval:

Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0.$ If the orthocenter of this triangle is at $(1, 1)$ then the equation of it's third side is:

Let the length of the latus rectum of an ellipse with its major axis along $x$ -axis and centre at the origin, be 8 . If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

Let $S={(x,y)\in {R}^{2}:\frac{{y}^{2}}{1+r}-\frac{{x}^{2}}{1-r}=1},$ where $r\neq \pm 1.$ Then $S$ represents:

Lines are drawn parallel to the line $4x-3y+2=0,$ at a distance $\frac{3}{5}$ units from the origin. Then which one of the following points lies on any of these lines?

Slope of a line passing through $P(2, 3)$ and intersecting the line $x+y=7$ at a distance of $4$ units from $P,$ is

Suppose that the points $(h, k), (1, 2)$ and $(-3, 4)$ lie on the line ${L}_{1}.$ If a line ${L}_{2}$ passing through the points $(h, k)$ and $(4, 3)$ is perpendicular to ${L}_{1},$ then $\frac{k}{h}$ equals:

The length of the chord of the parabola ${x}^{2}=4y$ having equation $x-\sqrt{2}y+4\sqrt{2}=0$ is

The locus of the centres of the circles, which touch the circle, ${x}^{2}+{y}^{2}=1$ externally, also touch the $y$-axis and lie in the first quadrant, is:

The sum of the squares of the lengths of the chords intercepted on the circle, ${x}^{2}+{y}^{2}=16,$ by the lines, $x+y=n, n\in N,$ where $N$ is the set of all natural numbers is:

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle ${x}^{2}+{y}^{2}=4$ and the $x$ -axis form a triangle. The area of this triangle (in square units) is:

Three circles of radii $a, b, c, (a<b<c)$ touch each other externally. If they have $x-$ axis as a common tangent, then:

Two circles with equal radii are intersecting at the points (0,1) and (0,-1) . The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0.$ If its diagonals intersect at $(2,4),$ then one of its vertex is:

Two vertices of a triangle are $(0,2)$ and $(4,3).$ If its orthocenter is at the origin, then its third vertex lies in which quadrant?