Coordinate Geometry PYQ — Page 15

If the variable line $3x+4y=\alpha$ lies between the two circles $(x-1{)}^{2}+(y-1{)}^{2}=1$ and $(x-9{)}^{2}+(y-1{)}^{2}=4,$ without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is

Let the circle $S:36{x}^{2}+36{y}^{2}-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle $S,$ then:

Consider a circle $C$ which touches the $y-$ axis at $(0,6)$ and cuts off an intercept $6\sqrt{5}$ on the $x-$ axis. Then the radius of the circle $C$ is equal to :

Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0).$ If the locus of the point $P$ is a circle of radius $r,$ then $4{r}^{2}$ (in the nearest integer) is equal to __________.

Choose the correct statement about two circles whose equations are given below: ${x}^{2}+{y}^{2}-10x-10y+41=0$ ${x}^{2}+{y}^{2}-22x-10y+137=0$

Let $A$ be the set of all points $(\alpha ,\beta )$ such that the area of triangle formed by the points $(5,6),$ $(3,2)$ and $(\alpha ,\beta )$ is $12$ square units. Then the least possible length of a line segment joining the origin to a point in $A,$ is :

Let the points of intersections of the lines $x-y+1=0,x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle $\mathrm{ABC}.$ Then the area of the triangle $\mathrm{ABC}$ is

The locus of the mid-point of the line segment joining the focus of the parabola ${y}^{2}=4ax$ to a moving point of the parabola, is another parabola whose directrix is:

If the points of intersection of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{{b}^{2}}=1$ and the circle ${x}^{2}+{y}^{2}=4b,b>4$ lie on the curve ${y}^{2}=3{x}^{2}$, then $b$ is equal to :

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\Delta ABC$, then $(R+r)$ is equal to :

If the curve ${x}^{2}+2{y}^{2}=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is

The equation of the circle with centre (1, -2) and radius 3 is:

A hyperbola passes through the foci of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

Consider a triangle having vertices $A(-2,3),B(1,9)$ and $C(3,8).$ If a line $L$ passing through the circum-centre of triangle $ABC,$ bisects line $BC,$ and intersects $y$-axis at point $(0,\frac{\alpha }{2}),$ then the value of real number $\alpha$ is _______.

The point $P(a,b)$ undergoes the following three transformations successively: $(a)$ reflection about the line $y=x.$ $(b)$ translation through $2$ units along the positive direction of $x-$ axis. $(c)$ rotation through angle $\frac{\pi }{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$, then the value of $2a+b$ is equal to:

Two tangents are drawn from the point $P(-1,1)$ to the circle ${x}^{2}+{y}^{2}-2x-6y+6=0.$ If these tangents touch the circle at points $A$ and $B,$ and if $D$ is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle $ABD$ is equal to:

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, ${x}^{2}+{y}^{2}=1$ is a circle of radius $r$, then $r$ is equal to

The locus of the point of intersection of the lines $(\sqrt{3})kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4(\sqrt{3})k=0$ is a conic, whose eccentricity is

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y-$ axis and then passes through the point $(5,3).$ If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}},$ then the equation of the other directrix can be:

Let $A(a,0),B(b,2b+1)$ and $C(0,b),b\neq 0,|b|\neq 1$, be points such that the area of triangle $ABC$ is $1$ sq. unit, then the sum of all possible values of $a$ is:

Let $\mathrm{tan}\alpha ,\mathrm{tan}\beta$ and $\mathrm{tan}\gamma ;\alpha ,\beta ,\gamma \neq \frac{(2n-1)\pi }{2},n\in N$ be the slopes of the three line segments $OA,OB$ and $OC,$ respectively, where $O$ is origin. If circumcentre of $\Delta ABC$ coincides with origin and its orthocentre lies on $y$-axis, then the value of ${(\frac{\mathrm{cos}3\alpha +\mathrm{cos}3\beta +\mathrm{cos}3\gamma }{\mathrm{cos}\alpha \cdot \mathrm{cos}\beta \cdot \mathrm{cos}\gamma })}^{2}$ is equal to :

Let ${r}_{1}$ and ${r}_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle ${x}^{2}+{y}^{2}+2x+4y-4=0$. If $\frac{{r}_{1}}{{r}_{2}}=a+b\sqrt{2},$ then $a+b$ is equal to:

The minimum distance between any two points ${P}_{1}$ and ${P}_{2}$ while considering point ${P}_{1}$ on one circle and point ${P}_{2}$ on the other circle for the given circles' equations ${x}^{2}+{y}^{2}-10x-10y+41=0$ ${x}^{2}+{y}^{2}-24x-10y+160=0$ is ________

Let $A={(x,y)\in R\times R\mid 2{x}^{2}+2{y}^{2}-2x-2y=1}$ $B={(x,y)\in R\times R\mid 4{x}^{2}+4{y}^{2}-16y+7=0}$ and $C={(x,y)\in R\times R\mid {x}^{2}+{y}^{2}-4x-2y+5\leq {r}^{2}}$. Then the minimum value of $|r|$ such that $A\cup B\subseteq C$ is equal to