Coordinate Geometry PYQ — Page 18

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 :

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 :

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:

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

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

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:

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?

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:

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 :

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:

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

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 $S$ 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:

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 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:

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 :

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:

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:

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 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 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:

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: