Straight Lines

In a triangle$ABC$, if $\cos\,A+2\cos\,B+\cos\,C=2$and the lengths of the sides opposite to the angles$A$and $C$ are $3$ and $7$ respectively, then $\cos\,A-\cos\,C$is equal to

If $4 x-5 y+33=0$ and $20 x-9 y=107$ are two lines of regression, then what are the values of $\bar{x}$ and $\bar{y}$ respectively?

Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS=70^{\circ},\angle PQS=15^{\circ}$ and $\angle PRS=40^{\circ}$. If $\angle RPS=\theta^{\circ},PQ=\alpha$ and $PS=\beta$, then the interval(s)…

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is

Paragraph: Let $F_{1}\left(x_{1}, 0\right)$ and $F_{2}\left(x_{2}, 0\right)$, for $x_{1}0$, be the foci of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .$ Suppose a parabola having vertex at the origin and focus at…

Paragraph: Let $S=S_{1} \cap S_{2} \cap S_{3}$, where $S_{1}=\{z \in \mathbb{C}:|z| 0\right\}$ and $S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$. Question: Area of $S=$

A straight line $L$ through the point $(3,-2)$ is inclined at an angle $60^{\circ}$ to the line $\sqrt{3} x+y=1$. If $L$ also intersects the $X$-axis, then the equation of $L$ is

Paragraph: Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.Question: The orthocentre of the triangle $P A B$ is

The locus of the orthocentre of the triangle formed by the lines $(1+p) x-p y+p(1+p)=0$, $(1+q) x-q y+q(1+q)=0$ and $y=0$, where $p \neq q$, is

Paragraph: A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation…

A straight line through the vertex $P$ of a $\triangle P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the $\triangle P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then

Lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$. Statement I The ratio $P R: R Q$…