JEE Main 2019 — Coordinate Geometry

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 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 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:

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:

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 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 :

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 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:

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:

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 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:

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:

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 $P(4,-4)$ 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 :

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

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:

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 :

A hyperbola has its centre at the origin, passes through the point $(4, 2)$ and has transverse axis of length $4$ along the $x-axis.$ Then the eccentricity of the hyperbola is:

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:

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

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:

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:

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?

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

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:

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?

A circle touching the $x-$ axis at $(3, 0)$ and making an intercept of length $8$ on the $y-$ axis passes through the point:

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:

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?

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

A circle cuts a chord of length 4 a on the $x$ -axis and passes through a point on the $y$ -axis, distant $2 \mathrm{~b}$ from the origin. Then the locus of the centre of this circle, 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:

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:

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

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

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?

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 ?

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 :

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:

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 :

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.

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:

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:

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

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:

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:

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 :

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: