Distance Formula in Cartesian Plane

Q. Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P{C}_{1}Q{C}_2$ is :

1. $4$
2. $6$
3. $8$
4. $9$

Q. The locus of a point, which moves such that the sum of squares of its distance from the points $(0,0),(1,0),(0,1),(1,1)$ is $18$ units, is a circle of diameter $d.$ Then $d^2$ is equal to

Q. Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0).$ If the locus of the point $P$ is a circle of radius $r,$ then $4r^2$ is equal to

Q. $L=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]=P+Q$, $P$ is a symmetric matrix, $Q$ is a skew-symmetric matrix then $P$ is equal to

1. $\left[\begin{array}{ccc}3& 5& 6\\ 5& 6& 4\\ 9& 4& 3\end{array}\right]$
2. $\left[\begin{array}{ccc}2& 3.5& 3\\ 3.5& 1& 2\\ 3& 2& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}6& 5& 4\\ 3& 6& 3\\ 5& 2& 5\end{array}\right]$
4. $\left[\begin{array}{ccc}6& 5& 4\\ 4& 5& 3\\ 3& 4& 3\end{array}\right]$

Q. Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0).$ If the locus of the point $P$ is a circle of radius $r,$ then $4r^2$ is equal to

Q. Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P{C}_{1}Q{C}_2$ is :

1. $4$
2. $6$
3. $8$
4. $9$

Q. Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P{C}_{1}Q{C}_2$ is :

1. $4$
2. $6$
3. $8$
4. $9$

Q. If $a,b,c$ are $p^{th},q^{th}$ and $r^{th}$ terms respectively of a H.P., then value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$ is

1. $0$
2. $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$
3. none of these
4. $p+q+r$

Q. The line $y=5-2x$ intersects the curve $3y^2=8x^2-6$ at two points, P and Q. Find the length of PQ and the midpoint of PQ.

Q. Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0).$ If the locus of the point $P$ is a circle of radius $r,$ then $4r^2$ is equal to

Q. If $a,b,c,d$ are positive and are the $p^{th},q^{th},r^{th}$ terms respectively of a G.P. show without expanding that,
$\left|\begin{array}{ccc}loga& p& 1\\ logb& q& 1\\ logc& r& 1\end{array}\right|=0$

Q. Let $P$ and $Q$ be 2*2 matrices. Consider the statements.
$PQ=0\Rightarrow 1)P=0$ or $Q=0$ or $both$
2) $PQ=I_2\Rightarrow P=Q^{-1}$
3) $(P+Q{)}^2=P^2+2PQ+Q^2$

1. $1$ and $2$ are false but $3$ is true
2. $1$ and $3$ false and $2$ is true
3. All are true
4. All are false

Q. Find the value of $c$ if the points $(2,0),(0,1),(4,5)$ and $(0,c)$ are concylic.

Q. If $p,q,r$ are positive integers, and
$\Delta =12\left|\begin{array}{ccc}{}^p{C}_1& {}^p{C}_2& {}^p{C}_3\\ {}^q{C}_1& {}^q{C}_2& {}^q{C}_3\\ {}^r{C}_1& {}^r{C}_2& {}^r{C}_3\end{array}\right|$
then

1. $\Delta$ is an even integer
2. None of these
3. $\Delta$ is divisible by 12
4. $\Delta$ is an rational number

Q. $(1-x^2)\frac{dy}{dx}+2xy=x(1-x^2{)}^{1/2}$.

1. $y=c(1+y^2)+\sqrt{1-x^2}$
2. $x=c(1-y^2)+\sqrt{1-y^2}$
3. $y=c(1-x^2)+\sqrt{1-x^2}$
4. $y=c(1-x^2)+\sqrt{1-y^2}$