Coordinate Geometry PYQ — Page 18

A hyperbola having the transverse axis of length, $\sqrt{2}$ has the same foci as that of the ellipse, $3{x}^{2}+4{y}^{2}=12$ then this hyperbola does not pass through which of the following points?

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola ${x}^{2}=4y$ internally in the ratio $1:2$ is:

If ${e}_{1}$ and ${e}_{2}$ are the eccentricities of the ellipse $\frac{{x}^{2}}{18}+\frac{{y}^{2}}{4}=1$ and the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$ respectively and $({e}_{1},{e}_{2})$ is a point on the ellipse $15{x}^{2}+3{y}^{2}=k$ , then the value of $k$ is equal to

The locus of the centre of a circle which touches the line x=1 and passes through the origin is:

Let ${e}_{1}$ and ${e}_{2}$ be the eccentricities of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{{b}^{2}}=1(b<5)$ and the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{{b}^{2}}=1$ respectively satisfying ${e}_{1}{e}_{2}=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to:

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$, to the line $x=y$, 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:

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 $S$ 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:

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 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:

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: