Coordinate Geometry PYQ — Page 14

Let $x=2t,y=\frac{{t}^{2}}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $SA\perp BA$, where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of the $\Delta SAB$, then $\underset{t\rightarrow 1}{\mathrm{lim}}k$ is equal to

Let $A(\frac{3}{\sqrt{a}},\sqrt{a}),a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\mathrm{cos}\theta ,a\mathrm{sin}\theta )$, is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is $12$ square units, then $a$ is equal to _____

Let the point $P(\alpha ,\beta )$ be at a unit distance from each of the two lines ${L}_{1}:3x-4y+12=0$, and ${L}_{2}:8x+6y+11=0$. If $P$ lies below ${L}_{1}$ and above ${L}_{2}$, then $100(\alpha +\beta )$ is equal to

The locus of the mid-point of the line segment joining the point $(4,3)$ and the points on the ellipse ${x}^{2}+2{y}^{2}=4$ is an ellipse with eccentricity

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point $B$. If the length of the line segment $AB$ is $\frac{\sqrt{29}}{3}$, then $B$ also lies on the line

Let $S={(x,y)\in \mathbb{N}\times \mathbb{N}:9{(x-3)}^{2}+16{(y-4)}^{2}\leq 144}$ and $T={(x,y)\in \mathbb{R}\times \mathbb{R}:{(x-7)}^{2}+{(y-4)}^{2}\leq 36}$ The $n(S\cap T)$ is equal to ______.

The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is $(\frac{7}{3},\frac{7}{3})$ is:

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line ${L}_{1}=4x+3y+2$ passes through the centre $P$ of the circle $C$ and intersects the line ${L}_{2}:3x-4y-11=0$ at $Q$. The line ${L}_{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5x-12y+51=0$ is

Let $C$ be the centre of the circle ${x}^{2}+{y}^{2}-x+2y=\frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi }{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in ${\mathrm{unit}}^{2}$) is

Let the tangents at two points $A$ and $B$ on the circle ${x}^{2}+{y}^{2}-4x+3=0$ meet at origin $O(0,0)$. Then the area of the triangle of $OAB$ is

Let $P:{y}^{2}=4ax,a>0$ be a parabola with focus $S$.Let the tangents to the parabola $P$ make an angle of $\frac{\pi }{4}$ with the line $y=3x+5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A,B$ and $S$ are collinear is:

Let the lines $y+2x=\sqrt{11}+7\sqrt{7}$ and $2y+x=2\sqrt{11}+6\sqrt{7}$ be normal to a circle $C:{(x-h)}^{2}+{(y-k)}^{2}={r}^{2}$. If the line $\sqrt{11}y-3x=\frac{5\sqrt{77}}{3}+11$ is tangent to the circle $C$, then the value of ${(5h-8k)}^{2}+5{r}^{2}$ is equal to ______.

The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0,x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the following is NOT true

Let $A(\alpha ,-2),B(\alpha ,6)$ and $C(\frac{\alpha }{4},-2)$ be vertices of a $\Delta ABC$. If $(5,\frac{\alpha }{4})$ is the circumcentre of $\Delta ABC$, then which of the following is NOT correct about $\Delta ABC$

If the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2\sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse is

Let the circumcentre of a triangle with vertices $A(a,3),B(b,5)$ and $C(a,b),ab>0$ be $P(1,1)$. If the line $AP$ intersects the line $BC$ at the point $Q({k}_{1},{k}_{2})$, then ${k}_{1}+{k}_{2}$ is equal to

For $t\in (0,2\pi )$, if $ABC$ is an equilateral triangle with vertices $A(\mathrm{sin}t,-\mathrm{cos}t),B(\mathrm{cos}t,\mathrm{sin}t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $(1,\frac{1}{3})$, then $({a}^{2}-{b}^{2})$ is equal to

A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is $14$. Let $f(x,y)=0$ be the locus of $P$, which intersects the $x$-axis at the points $A,B$ and the $y$-axis at the point $C,D$. Then the area of the quadrilateral $ACBD$ is equal to

If the circle ${x}^{2}+{y}^{2}-2gx+6y-19c=0,g,c\in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on $x$-axis is

Let the hyperbola $H:\frac{{x}^{2}}{{a}^{2}}-{y}^{2}=1$ and the ellipse $E:3{x}^{2}+4{y}^{2}=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If ${e}_{H}$ and ${e}_{E}$ are the eccentricities of $H$ and $E$ respectively, then the value of $12({e}_{H}^{2}+{e}_{E}^{2})$ is equal to _____.

Let a triangle be bounded by the lines ${L}_{1}:2x+5y=10$; ${L}_{2}:-4x+3y=12$ and the line ${L}_{3}$, which passes through the point $P(2,3)$, intersect ${L}_{2}$ at $A$ and ${L}_{1}$ at $B$. If the point $P$ divides the line-segment $AB$, internally in the ratio $1:3$, then the area of the triangle is equal to

Let the tangent to the circle ${C}_{1}:{x}^{2}+{y}^{2}=2$ at the point $M(-1,1)$ intersect the circle ${C}_{2}$ : ${(x-3)}^{2}+{(y-2)}^{2}=5$, at two distinct points $A$ and $B$. If the tangents to ${C}_{2}$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $ANB$ is equal to

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16{x}^{2}-9{y}^{2}+32x+36y-164=0$ and its foci is

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y-$ axis and then passes through the point $(5,3).$ If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}},$ then the equation of the other directrix can be: