Coordinate Geometry PYQ — Page 14

Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle ${(x-5)}^{2}+{(y-1)}^{2}=\frac{13}{2}$, then ${r}^{2}$ is equal to

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 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

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:

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 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 tangents at the points $P$ and $Q$ on the ellipse $\frac{{x}^{2}}{2}+\frac{{y}^{2}}{4}=1$ meet at the point $R(\sqrt{2},2\sqrt{2}-2)$. If $S$ is the focus of the ellipse on its negative major axis, then $S{P}^{2}+S{Q}^{2}$ is equal to

If one of the diameters of the circle ${x}^{2}+{y}^{2}-2\sqrt{2}x$ $-6\sqrt{2}y+14=0$ is a chord of the circle ${(x-2\sqrt{2})}^{2}$ $+{(y-2\sqrt{2})}^{2}={r}^{2}$, then the value of ${r}^{2}$ 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

A rectangle $R$ with end points of the one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is _____.

Let the foci of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{7}=1$ and the hyperbola $\frac{{x}^{2}}{144}-\frac{{y}^{2}}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola 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

If the line $x-1=0$, is a directrix of the hyperbola $k{x}^{2}-{y}^{2}=6$, then the hyperbola passes through the point

Let $AB$ be a chord of length $12$ of the circle ${(x-2)}^{2}+{(y+1)}^{2}=\frac{169}{4}$ If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$, then five times the distance of point $P$ from chord $AB$ 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 locus of the centre $(\alpha ,\beta ),\beta >0$, of the circle which touches the circle ${x}^{2}+{(y-1)}^{2}=1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y=4$ 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 _____.

In an isosceles triangle $ABC$, the vertex $A$ is $(6,1)$ and the equation of the base $BC$ is $2x+y=4$. Let the point $B$ lie on the line $x+3y=7$. If $(\alpha ,\beta )$ is the centroid $\Delta ABC$, then $15(\alpha +\beta )$ 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

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

Consider a hyperbola $H:{x}^{2}-2{y}^{2}=4$. Let the tangent at a point $P(4,\sqrt{6})$ meet the $x$-axis at $Q$ and latus rectum at $R({x}_{1},{y}_{1}),{x}_{1}>0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\Delta QFR$ (in sq. units) is equal to