Coordinate Geometry PYQ — Page 9

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2 q-p)^2$ is equal to ________

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$. If $\mathrm{e}$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8 \mathrm{e}^2+l^2$ is equal to.

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

Let $C:{x}^{2}+{y}^{2}=4$ and ${C}^{'}:{x}^{2}+{y}^{2}-4\lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $R-[a,b]$, then the point $(8a+12,16b-20)$ lies on the curve:

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 :

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the circle $\mathrm{C}$ at the points $\mathrm{P}$ and $\mathrm{Q}$. Let $\mathrm{MN}$ be a chord of $\mathrm{C}$ of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord $\mathrm{MN}$ is

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 :

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 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 $P$ be a point on the ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$. Let the line passing through $P$ and parallel to $y-$axis meet the circle ${x}^{2}+{y}^{2}=9$ at point $Q$ such that $P\text{and}Q$ are on the same side of the $x-$axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse, 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 _________

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

Let $(5,\frac{a}{4})$, be the circumcenter of a triangle with vertices $A(a,-2),B(a,6)$ and $C(\frac{a}{4},-2)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha +\beta +\gamma$ is

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 $A$ be the point of intersection of the lines $3x+2y=14,5x-y=6$ and $B$ be the point of intersection of the lines $4x+3y=8,6x+y=5$. The distance of the point $P(5,-2)$ from the line $AB$ is

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 ${x}^{2}-{y}^{2}+2hxy+2gx+2fy+c=0$ 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

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 $A(a,b),B(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

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 _______.

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 :

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 _________.