Let the domains of the functions $f(x)={\log }_4{\log }_3{\log }_7\left(8-{\log }_2\left(x^2+4x+5\right)\right)$ and $\mathrm{g}(x)={\sin }^{-1}\left(\frac{7x+10}{x-2}\right)$ be $(\alpha ,\beta )$ and $[\gamma ,\delta ]$, respectively. Then ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$ is equal to :

Let $f,g:(1,\infty )\to \mathbb{R}$ be defined as $f(x)=\frac{2x+3}{5x+2}$ and $g(x)=\frac{2-3x}{1-x}$. If the range of the function fog: $[2,4]\to \mathbb{R}$ is $[\alpha ,\beta ]$, then $\frac{1}{\beta -\alpha }$ is equal to

Let $f$ be a function such that $f(x)+3f\left(\frac{24}{x}\right)=4x,x\ne 0$. Then $f(3)+f(8)$ is equal to

If the domain of the function $f(x)={\log }_7\left(1-{\log }_4\left(x^2-9x+18\right)\right)$ is $(\alpha ,\beta )\cup (\gamma ,o)$, then $\alpha +\beta +\gamma +\hat{o}$ is 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 intersect at the points A and B, then (AB)2 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 $△PQR$ 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 the line $3 x-2 y+12=0$ intersects the parabola $4y=3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let the shortest distance from $(a, 0), a>0$, to the parabola $y^2=4x$ 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 :

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

Let ABCD be a trapezium whose vertices lie on the parabola ${\mathrm{y}}^2=4\mathrm{x}$. Let the sides AD and BC 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

Let the focal chord PQ of the parabola $y^2=4x$ 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 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

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

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

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$, parabola $P$ and the $x$-axis, 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 $(1, k)$ lies on the parabola, then a possible value of k is :

Let $f(x)=\underset{0}{\overset{x}{\int }}\mathrm{t}\left({\mathrm{t}}^2-9\mathrm{t}+20\right)\mathrm{dt},1\le x\le 5$. If the range of $f$ is $[\alpha ,\beta ]$, then $4(\alpha +\beta )$ equals :

The integral $80\underset{0}{\overset{\frac{\pi }{4}}{\int }}\left(\frac{\sin \theta +\cos \theta }{9+16\sin 2\theta }\right)d\theta$ is equal to :