JEE Main 2024 — Coordinate Geometry

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

Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci on the $\mathrm{x}$-axis. Let $\mathrm{C}_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $\mathrm{C}_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points $({a}^{2},a+1)$ lie in $R$, is :

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

Consider a triangle $\mathrm{ABC}$ having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle A B C=\frac{\pi}{6}$ and $\angle B A C=\frac{2 \pi}{3}$. If the points $\mathrm{B}$ and $\mathrm{C}$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ is equal to ________

For $0<\theta <\pi /2$, if the eccentricity of the hyperbola ${x}^{2}-{y}^{2}{\mathrm{cosec}}^{2}\theta =5$ is $\sqrt{7}$ times eccentricity of the ellipse ${x}^{2}{\mathrm{cosec}}^{2}\theta +{y}^{2}=5$, then the value of $\theta$ is:

Let the foci and length of the latus rectum of an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$, $a>b$ be $(\pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{{x}^{2}}{{b}^{2}}-\frac{{y}^{2}}{{a}^{2}{b}^{2}}=1$ equals

Let a ray of light passing through the point $(3,10)$ reflects on the line $2 x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the incident ray is $a x+b y+1=0$, then $a^2+b^2+3 a b$ is equal to_________

If the orthocentre of the triangle formed by the lines $2 x+3 y-1=0, x+2 y-1=0$ and $a x+b y-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$, then the value of $|a-b|$ is_______

The equations of two sides $\mathrm{AB}$ and $\mathrm{AC}$ of a triangle $\mathrm{ABC}$ are $4 x+y=14$ and $3 x-2 y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side $\mathrm{BC}$ internally in the ratio $2: 1$. the equation of the side $\mathrm{BC}$ is

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines ${L}_{1}$ and ${L}_{2}$ passing through the origin. The tangent of an angle between the lines ${L}_{1}$ and ${L}_{2}$ is :

A square is inscribed in the circle $x^2+y^2-10 x-6 y+30=0$. One side of this square is parallel to $y=x+3$. If $\left(x_i, y_i\right)$ are the vertices of the square, then $\mathbf{\Sigma}\left(x_i^2+y_i^2\right)$ is equal to:

Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $\mathrm{C}$ from the point $(5,5)$ is :

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$, then $12 d$ equals ______

Let the length of the focal chord PQ of the parabola $y^2=12 x$ be 15 units. If the distance of $\mathrm{PQ}$ from the origin is $\mathrm{p}$, then $10 \mathrm{p}^2$ is equal to _______

Let PQ be a chord of the parabola $y^2=12 x$ and the midpoint of PQ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and Q?

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose $\text { length of semi major axis is } x \text { and semi minor axis is } y$, eccentricity is $\mathrm{e}$ and the length of the latus rectum is $l$, then $81\left(e^ 4+l^2\right)$ is equal to

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $a^2+2 b+3 c+4 d+e$ is equal to :

A ray of light coming from the point $P(1,2)$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h, k)$ is such that PQRS is a parallelogram, then $h k^2$ is equal to :

Let $C$ be the circle of minimum area touching the parabola $y=6-x^2$ and the lines $y=\sqrt{3}|x|$. Then, which one of the following points lies on the circle $C$ ?

Let the locus of the mid points of the chords of circle ${x}^{2}+{(y-1)}^{2}=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then, the length of $PQ$ is:

If the sum of squares of all real values of $\alpha$, for which the lines $2x-y+3=0,6x+3y+1=0$ and $\alpha x+2y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is ________.

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 :

If the shortest distance of the parabola ${y}^{2}=4x$ from the centre of the circle ${x}^{2}+{y}^{2}-4x-16y+64=0$ is $d$, then ${d}^{2}$ is equal to :

Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is 10 . If the locus of $\mathrm{P}$ is $\mathrm{a} x+\mathrm{b} y=15$, then $5 \mathrm{a}+2 \mathrm{~b}$ is :

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $\mathrm{C}_1$ and $\mathrm{C}_2$ intersects the $y$-axis at the point $\mathrm{P}$, then the square of the distance of $\mathrm{P}$ from the centre of $\mathrm{C}_1$ is :

Consider a circle $(x-{\alpha }^{2})+(y-{\beta }^{2})=50$, where $\alpha ,\beta >0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4\sqrt{2}$ , then $(\alpha +\beta {)}^{2}$ is equal to _______.

The length of the chord of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$, whose mid point is $(1,\frac{2}{5})$, is equal to:

Let $ABC$ be an isosceles triangle in which $A$ is at $(-1,0),\angle A=\frac{2\pi }{3},AB=AC$ and $B$ is on the positive $x-$axis. If $BC=4\sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha ,\beta ),$ then $\frac{{\beta }^{4}}{{\alpha }^{2}}$ is:

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is $56$ more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is

Let $\alpha ,\beta ,\gamma ,\delta \in Z$ and let $A(\alpha ,\beta ),B(1,0),C(\gamma ,\delta )$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha +\beta +\gamma +\delta )$ is equal to

The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to_________

Let the line $L:\sqrt{2}x+y=\alpha$ pass through the point of the intersection $P$(in the first quadrant)of the circle ${x}^{2}+{y}^{2}=3$ and the parabola ${x}^{2}=2y$. Let the line $L$ touch two circles ${C}_{1}$ and ${C}_{2}$ of equal radius $2\sqrt{3}$. If the centres ${Q}_{1}$ and ${Q}_{2}$ of the circles ${C}_{1}$ and ${C}_{2}$ lie on the $y-$axis, then the square of the area of the triangle $P{Q}_{1}{Q}_{2}$ is equal to _________.

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to

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:

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $(-\frac{22}{7},-4)$ and $(-\frac{1}{7},3)$ intersects the circle at only one point $P(\alpha ,\beta )$. Then $17\beta -\alpha$ is equal to

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

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 _________.

Let $\mathrm{S}$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x$-axis. Let $\mathrm{C}$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to___________

Suppose $A B$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m} < \sqrt{3}$. If the distance of the chord $\mathrm{AB}$ from the origin is $\mathrm{d}$, then $l \mathrm{~d}^2$ is equal to _______

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 $P(\alpha ,\beta )$ be a point on the parabola ${y}^{2}=4x$. If $P$ also lies on the chord of the parabola ${x}^{2}=8y$ whose mid point is $(1,\frac{5}{4})$, then $(\alpha -28)(\beta -8)$ 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

If one of the diameters of the circle ${x}^{2}+{y}^{2}-10x+4y+13=0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle $C$ is

Let $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ is:

A line passing through the point $A(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15^{\circ}$ in the clockwise direction, then its equation in the new position is

If the line segment joining the points $(5,2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

If the image of the point $(-4,5)$ in the line $x+2 y=2$ lies on the circle $(x+4)^2+(y-3)^2=r^2$, then $\mathrm{r}$ is equal to:

Let ${e}_{1}$ be the eccentricity of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ and ${e}_{2}$ be the eccentricity of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$, which passes through the foci of the hyperbola. If ${e}_{1}{e}_{2}=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is :

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

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is

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 __________.

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $\mathrm{A}(5,-2), \mathrm{B}(8,3)$ and $\mathrm{C}(\mathrm{h}, \mathrm{k})$, then the point $C$ lies on the circle: