JEE Main 2025 — Coordinate Geometry

Let $\mathrm{H}_1: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ and $\mathrm{H}_2:-\frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their ecentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 \mathrm{e}_2^2$ is equal to ________.

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{\mathrm{x}^2}{\mathrm{~b}^2}+\frac{\mathrm{y}^2}{25}=1 \quad$ and the hyperbola $\quad \frac{\mathrm{x}^2}{16}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$, respectively. If $\mathrm{b} \lt 5$ and $\mathrm{e}_1 \mathrm{e}_2=1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

Let $\mathrm{E}_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $\mathrm{E}_1$ 's are constructed such that their centres and eccentricities are same as that of $E_1$, and the length of minor axis of $E_i$ is the length of major axis of $E_{i+1}(i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi}\left(\sum_{i=1}^{\infty} A_i\right)$, is equal to

The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is

Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ be $8 \sqrt{\frac{5}{3}}$. If for $H$, the length of the latus rectum is $l$ and the product of the focal distances of the point P is m , then $9 l^2+6 \mathrm{~m}$ is equal to :-

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1: 2 \mathrm{x}+\mathrm{y}+6=0$ and $\mathrm{L}_2: 4 \mathrm{x}+2 \mathrm{y}-\mathrm{p}=0, \mathrm{p} \gt 0$, at the points $A$ and $B$, respectively. If $A B=\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point A on the line $L_2$ is $M$, then $\frac{A M}{B M}$ is equal to

Let a be the length of a side of a square OABC with $O$ being the origin. Its side OA makes an acute angle $\alpha$ with the positive $x$-axis and the equations of its diagonals are $(\sqrt{3}+1) x+(\sqrt{3}-1) y=0$ and $(\sqrt{3}-1) x-(\sqrt{3}+1) y+8 \sqrt{3}=0$. Then $\mathrm{a}^2$ is equal to

Let the three sides of a triangle are on the lines $4 x-7 y+10=0, x+y=5$ and $7 x+4 y=15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x=0, y=0$ and $x+y=1$ is

If the orthocentre of the triangle formed by the lines $y=x+1, y=4 x-8$ and $y=m x+c$ is at $(3,-1)$, then $\mathrm{m}-\mathrm{c}$ is :

Let the equation $\mathrm{x}(\mathrm{x}+2)(12-\mathrm{k})=2$ have equal roots. Then the distance of the point $\left(\mathrm{k}, \frac{\mathrm{k}}{2}\right)$ from the line $3 x+4 y+5=0$ is

Let the lines $3 x-4 y-\alpha=0,8 x-11 y-33=0$, and $2 x-3 y+\lambda=0$ be concurrent. If the image of the point $(1,2)$ in the line $2 x-3 y+\lambda=0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha \lambda|$ is equal to

Let circle $C$ be the image of $x^2+y^2-2 x+4 y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta \lt 4$, lies on $C$ such that the length of the are $A B$ is $(1 / 6)^{\text {th }}$ of the perimeter of $C$, then $\beta-\sqrt{3} \alpha$ is equal to

Let $r$ be the radius of the circle, which touches x -axis at point $(\mathrm{a}, 0), \mathrm{a} \lt 0$ and the parabola $\mathrm{y}^2=9 \mathrm{x}$ at the point $(4,6)$. Then $r$ is equal to ________

Let the focal chord $P Q$ of the parabola $y^2=4 x$ make an angle of $60^{\circ}$ with the positive $x$-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5 \alpha^2$ is equal to :

Let $A B C D$ be a trapezium whose vertices lie on the parabola $y^2=4 x$. Let the sides $A D$ and $B C$ of the trapezium be parallel to y -axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $A B C D$ is

The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2, \lambda_1 \lt \lambda_2$, then $12 \lambda_1+29 \lambda_2$ is equal to $\qquad$

Let C be the circle $\mathrm{x}^2+(\mathrm{y}-1)^2=2, \mathrm{E}_1$ and $\mathrm{E}_2$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x+y=3$ touch the curves $C$, $E_1$ and $E_2$ at $P\left(x_1, y_1\right), Q\left(x_2, y_2\right)$ and $R\left(x_3, y_3\right)$ respectively. Given that $P$ is the mid-point of the line segment $Q R$ and $P Q=\frac{2 \sqrt{2}}{3}$, the value of $9\left(x_1 y_1+x_2 y_2+x_3 y_3\right)$ is equal to ______ .

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(S P . S^{\prime} \mathrm{P}\right)+$ $\max \left(\mathrm{SP} . \mathrm{S}^{\prime} \mathrm{P}\right)$ is equal to :

Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(\mathrm{a}\gt\mathrm{b})$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

Let the line \(x+y=1\) meet the circle \(x^2+y^2=4\) at the points A and B . If the line perpendicular to \(A B\) and passing through the mid point of the chord \(A B\) intersects the circle at \(C\) and \(D\), then the area of the quadrilateral ADBC is equal to :

Let the equation of the circle, which touches $x$-axis at the point $(a, 0), a\gt0$ and cuts off an intercept of length $b$ on $y$-axis be $x^2+y^2-\alpha x+\beta y+\gamma=0$. If the circle lies below $x$-axis, then the ordered pair $\left(2 a, b^2\right)$ is equal to

Let ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}=28,{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=56$ and ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}=70$. Let $\mathrm{A}(4 \cos t, 4 \sin t), \mathrm{B}(2 \sin t,-2 \cos \mathrm{t})$ and $C\left(3 r-n, r^2-n-1\right)$ be the vertices of a triangle $A B C$, where $t$ is a parameter. If $(3 x-1)^2+(3 y)^2$ $=\alpha$, is the locus of the centroid of triangle ABC , then $\alpha$ equals

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $\mathrm{E}: \frac{\mathrm{x}^2}{4^2}+\frac{\mathrm{y}^2}{3^2}=1$ at the points A and B . Let the line $\mathrm{y}=\mathrm{x}$ intersect E at the points C and $D$. Then the area of the quadrilateral $A B C D$ is equal to

Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points $P, Q$ and $R$, then the area of $\triangle P Q R$ is :

The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$, whose mid-point is $\left(1, \frac{1}{2}\right)$, is :

The distance between two points (x₁,y₁) and (x₂,y₂) in a plane is:

A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(\alpha, \beta)$, then $3 \beta-2 \alpha$ is equal to :

Let $A(6,8), B(10 \cos \alpha,-10 \sin \alpha)$ and $C(-10 \sin \alpha, 10 \cos \alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5 a-3 h+6 k+100 \sin 2 \alpha)$ is equal to ______ -.

Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x+y=11, x+2 y=16$ and $2 x+3 y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $\triangle \mathrm{PQR}$ is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be 10 . If its eccentricity is the minimum value of the function $f(\mathrm{t})=\mathrm{t}^2+\mathrm{t}+\frac{11}{12}$, $\mathrm{t} \in \mathbf{R}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to :

The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ______.

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having one of its focus at $\mathrm{P}(-3,0)$. If the latus ractum through its other focus subtends a right angle at P and $a^2 b^2=\alpha \sqrt{2}-\beta, \alpha, \beta \in \mathbb{N}$.

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

Let $y^2=12 x$ be the parabola and $S$ be its focus. Let PQ be a focal chord of the parabola such that $(\mathrm{SP})(\mathrm{SQ})=\frac{147}{4}$. Let C be the circle described taking PQ as a diameter. If the equation of a circle $C$ is $64 x^2+64 y^2-\alpha x-64 \sqrt{3} y=\beta$, then $\beta-\alpha$ is equal to ________.

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^2-y^2-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to _____. 

Let the ellipse \(\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\gt\mathrm{b}\) and \(\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1, \mathrm{~A} \lt \mathrm{B}\) have same eccentricity \(\frac{1}{\sqrt{3}}\). Let the product of their lengths of latus rectums be \(\frac{32}{\sqrt{3}}\), and the distance between the foci of \(E_1\) be 4. If \(E_1\) and \(E_2\) meet at \(A, B, C\) and \(D\), then the area of the quadrilateral \(A B C D\) equals :

If the midpoint of a chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ is $(\sqrt{2}, 4 / 3)$, and the length of the chord is $\frac{2 \sqrt{\alpha}}{3}$, then $\alpha$ is :

A line passing through the point $\mathrm{P}(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{\mathrm{x}^2}{36}+\frac{\mathrm{y}^2}{25}=1$ at $A$ and $B$ such that $(P A) .(P B)$ is maximum. Then $5\left(P A^2+P B^2\right)$ is equal to :

Let ABC be the triangle such that the equations of lines $A B$ and $A C$ be $3 y-x=2$ and $x+y=2$, respectively, and the points B and C lie on x -axis. If $P$ is the orthocentre of the triangle $A B C$, then the area of the triangle PBC is equal to

Let A and B be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If d denotes the distance between A and B , and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y^2=4 x$, then the value of $(a+d)$ is $\qquad$ -

Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3 x+2 y+2=0$. Then the length of the chord, of the circle $C$, whose mid-point is $(1,2)$, is :

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0$, then $\alpha+\beta+\gamma$ is equal to :

Let the ellipse $3 x^2+\mathrm{py}^2=4$ pass through the centre $C$ of the circle $x^2+y^2-2 x-4 y-11=0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point C on the ellipse. Then $6 f_1 f_2-r$ is equal to

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its latus-rectum is :

Let $\mathrm{P}(4,4 \sqrt{3})$ be a point on the parabola $y^2=4 \mathrm{a} x$ and PQ be a focal chord of the parabola. If M and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to :

Let $\mathrm{A}(4,-2), \mathrm{B}(1,1)$ and $\mathrm{C}(9,-3)$ be the vertices of a triangle $A B C$. Then the maximum area of the parallelogram AFDE , formed with vertices $\mathrm{D}, \mathrm{E}$ and F on the sides $\mathrm{BC}, \mathrm{CA}$ and AB of the triangle ABC respectively, is ______ .

Consider the lines $\mathrm{x}(3 \lambda+1)+\mathrm{y}(7 \lambda+2)=17 \lambda+5$, $\lambda$ being a parameter, all passing through a point P . One of these lines (say L) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(\mathrm{k}, 3 \mathrm{k})$ lie on a circle of radius r , then $10 \mathrm{k}+\mathrm{r}^2$ is equal to

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $A B$ subtends an angle equal to

Two parabolas have the same focus \((4,3)\) and their directrices are the \(x\)-axis and the \(y\)-axis, respectively. If these parabolas intersects at the points \(A\) and \(B\), then \((A B)^2\) is equal to :

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line AB , parabola P and the x -axis, is :-

Let the line $x+y=1$ meet the axes of $x$ and $y$ at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines $O B$ and $A B$, respectively. If the area of the triangle $A M N$ is $\frac{4}{9}$ of the area of the triangle $O A B$ and AN : NB $=\lambda: 1$, then the sum of all possible value(s) of is $\lambda$ :

Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\gt\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$. Let the distance between the foci of E and the foci of $H$ be $2 \sqrt{3}$. If $a-A=2$, and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to:

Let the product of the focal distances of the point $\mathrm{P}(4,2 \sqrt{3})$ on the hyperbola $\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ be 32 . Let the length of the conjugate axis of $H$ be $p$ and the length of its latus rectum be q . Then $\mathrm{p}^2+\mathrm{q}^2$ is equal to _______

The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is

Let $A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leq 4 \text { and }|\beta-5| \leq 6\}$ and $B=\left\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}: 16(\alpha-2)^2+9(\beta-6)^2 \leq 144\right\}$

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $A B$ internally in the ratio $2: 1$ is $9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0$, then $\alpha-\beta-\gamma$ is equal to :

Let one focus of the hyperbola $\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ be at $(\sqrt{10}, 0)$ and the corresponding directrix be $\mathrm{x}=\frac{9}{\sqrt{10}}$. If e and $l$ respectively are the eccentricity and the length of the latus rectum of H , then $9\left(\mathrm{e}^2+l\right)$ is equal to:

The centre of a circle C is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a \gt b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle $\mathrm{PF}_1 \mathrm{~F}_2$ is 30 and the length of the major axis of E is 17 , then the distance between the foci of E is :

The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $(3,1)$ is :

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be $(-5,0)$ and $5 x+9=0$, respectively. If the product of the focal distances of a point $(\alpha, 2 \sqrt{5})$ on the hyperbola is $p$, then $4 p$ is equal to

If $A$ and $B$ are the points of intersection of the circle $x^2+y^2-8 x=0$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ and a point P moves on the line $2 x-3 y+4=0$, then the centroid of $\triangle \mathrm{PAB}$ lies on the line :

A line passing through the point $\mathrm{P}(\mathrm{a}, \theta)$ makes an acute angle $\alpha$ with the positive x -axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha}{2}$ in the clock-wise direction. If in the new position, the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$, then the value of $3 a^2 \tan ^2 \alpha-2 \sqrt{3}$ is

Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}$, then $\alpha+\sqrt{2}(\sqrt{3}-1) \beta$ is equal to

Two equal sides of an isosceles triangle are along $-x+2 y=4$ and $x+y=4$. If m is the slope of its third side, then the sum, of all possible distinct values of $m$, is :

Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $( \pm 2,0)$. Let PQR be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $Q R$ of length 29 is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle PQR is :

Let \(A B C\) be a triangle formed by the lines \(7 x-6 y+3=0, x+2 y-31=0\) and \(9 x-2 y-19=0\). Let the point \((h, k)\) be the image of the centroid of \(\Delta A B C\) in the line \(3 x+6 y-53=0\). Then \(h^2+k^2+h k\) is equal to:

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P , whose abscissa is -2 , is

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, \mathrm{k})$ lies on the parabola, then a possible value of $k$ is :-

Let the area of the triangle formed by a straight Line L: $\mathrm{x}+\mathrm{by}+\mathrm{c}=0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $45^{\circ}$ with the positive x -axis, then the value of $\mathrm{b}^2+\mathrm{c}^2$ is:

Let the shortest distance from $(\mathrm{a}, 0), \mathrm{a}\gt0$, to the parabola $y^2=4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

Let the point $P$ of the focal chord $P Q$ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord PQ in the ratio $\mathrm{m}: \mathrm{n}$, $\operatorname{gcd}(m, n)=1$, then $m^2+n^2$ is equal to :

Let $C_1$ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let $\mathrm{C}_2$ be the circle with centre $(1,3)$ that touches $\mathrm{C}_1$ externally at the point $(\alpha, \beta)$. If $(\beta-\alpha)^2=\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to :

Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive x -axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2 \alpha^2+3 \beta^2$ is equal to ______