Coordinate Geometry PYQ — Page 19

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:

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 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 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

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:

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:

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?

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

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

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 :

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 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 $P(4,-4)$ 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 :

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:

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:

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?