Coordinate Geometry PYQ — Page 19

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 ${C}_{1}$ 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 $A(4,-4)$ 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:

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 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 ?

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

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 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 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 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 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 :

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

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?

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