Coordinate Geometry PYQ — Page 9

If $\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)$ and $\mathrm{D}(-1,-4,-2)$ are the vertices of a quadrilateral $A B C D$, then its area is :

Let $A(-2,-1),B(1,0),C(\alpha ,\beta )$ and $D(\gamma ,\delta )$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $|\alpha +\beta +\gamma +\delta |$ is equal to ______.

Let $A(\alpha ,0)$ and $B(0,\beta )$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x-$axis passes through $P$, then the length of the latus rectum of $E$ is equal to

The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}x-y+1=0$, is equal to

Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $x^2+y^2-3 x-2 y=0$ and the length of the latus rectum of the ellipse $x^2+9 y^2=k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

Let a circle passing through $(2,0)$ have its centre at the point $(h, k)$. Let $\left(x_c, y_c\right)$ be the point of intersection of the lines $3 x+5 y=1$ and $(2+c) x+5 c^2 y=1$. If $\mathrm{h}=\lim _{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim _{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}$, then the equation of the circle is :

Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment $\mathrm{AB}$ and the point $\mathrm{F}$ is on the diagonal $\mathrm{AC}$. Then the radius $\mathrm{r}$ of the circle passing through the point $\mathrm{F}$ and touching the line segments $\mathrm{BC}$ and $\mathrm{CD}$ satisfies:

Let a line perpendicular to the line $2 x-y=10$ touch the parabola $y^2=4(x-9)$ at the point $P$. The distance of the point $P$ from the centre of the circle $x^2+y^2-14 x-8 y+56=0$ is __________

Let $A, B$ and $C$ be three points on the parabola $y^2=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\frac{A M \cdot B N}{C D}\right)^2$ is equal to _________

If the circles $(x+1{)}^{2}+(y+2{)}^{2}={r}^{2}$ and ${x}^{2}+{y}^{2}-4x-4y+4=0$ intersect at exactly two distinct points, then

Let a variable line passing through the centre of the circle ${x}^{2}+{y}^{2}-16x-4y=0$, meet the positive co-ordinate axes at the point $A\text{and}B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to

If $A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)$ and $D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral $A B C D$, then its area is

Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

Four distinct points $(2k,3k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

Let the latus rectum of the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{{b}^{2}}=1$ subtend an angle of $\frac{\pi }{3}$ at the centre of the hyperbola. If ${b}^{2}$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then ${l}^{2}+{m}^{2}+{n}^{2}$ is equal to __________.

Let two straight lines drawn from the origin $\mathrm{O}$ intersect the line $3 x+4 y=12$ at the points $\mathrm{P}$ and $\mathrm{Q}$ such that $\triangle \mathrm{OPQ}$ is an isosceles triangle and $\angle \mathrm{POQ}=90^{\circ}$. If $l=\mathrm{OP}^2+\mathrm{PQ}^2+\mathrm{QO}^2$, then the greatest integer less than or equal to $l$ is :

If ${x}^{2}-{y}^{2}+2hxy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$, then the value of $g+c+h-f$ equals

Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

Let $P$ be a point on the hyperbola $H:\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is

If the foci of a hyperbola are same as that of the ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $(\sqrt{2},\frac{14}{3}\sqrt{\frac{2}{5}})$ on the hyperbola, is equal to

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6), \alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to

If the points of intersection of two distinct conics ${x}^{2}+{y}^{2}=4b$ and $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{{b}^{2}}=1$ lie on the curve ${y}^{2}=3{x}^{2}$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is _______.

Consider two circles ${C}_{1}:{x}^{2}+{y}^{2}=25$ and ${C}_{2}:(x-\alpha {)}^{2}+{y}^{2}=16$, where $\alpha \in (5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of ${C}_{1}$and ${C}_{2}$ be ${\mathrm{sin}}^{-1}(\frac{\sqrt{63}}{8})$. If the length of common chord of ${C}_{1}$ and ${C}_{2}$ is $\beta$, then the value of $(\alpha \beta {)}^{2}$ equals _________.

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2{x}^{2}+54$ at points $(x,y)$ and $(-x,y)$ where $y>0$ is :