Distance and section formulae

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 $\dfrac{\sqrt{29}}3$, then B also lies on the line :

Let A(1, 1), B(−4, 3), C(−2, −5) be vertices of a triangle ABC, P be a point on side BC, and $\Delta_1$ and $\Delta_2$ be the areas of triangles APB and ABC, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines AP, AC and the $x$-axis is

For $t\in(0,2\pi)$, if ABC is an equilateral triangle with vertices $A(\sin t,-\cos t)$, $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\dfrac13\right)$, then $\left(a^2-b^2\right)$ is equal to :

Let the mirror image of a circle $c_1:\ x^2+y^2-2x-6y+\alpha=0$ in line $y=x+1$ be $c_2:\ 5x^2+5y^2+10gx+10fy+38=0$. If r is the radius of circle $c_2$, then $\alpha+6r^2$ is equal to _______.

Let $E$ be the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. For any three distinct points $P$, $Q$ and $Q'$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$, and $M(P, Q')$ be the mid-point of the line segment joining $P$ and $Q'$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q')$, as $P$, $Q$ and $Q'$ vary on $E$, is ____.

If the tangent to the curve $y=x^3$ at the point P(t, t$^3$) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

Let k be an integer such that the triangle with vertices $(k,-3k)$, $(5,k)$ and $(-k,2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point :

A square, of each side 2, lies above the $x$-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the $x$-axis, then the sum of the $x$-coordinates of the vertices of the square is :

A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :

The points $\left(0,\dfrac83\right)$, (1, 3) and (82, 30) :

Let PS be the median of the triangle with vertices $P(2,2)$, $Q(6,-1)$ and $R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to PS is :

Given three points P, Q, R with P(5, 3) and R lies on the $x$-axis. If equation of RQ is $x-2y=2$ and PQ is parallel to the $x$-axis, then the centroid of $\Delta PQR$ lies on the line :

Let $A(2,3,5)$, $B(-1,3,2)$ and $C(\lambda,5,\mu)$ be the vertices of a $\Delta ABC$. If the median through A is equally inclined to the coordinate axes, then :

If a line intercepted between the coordinate axes is trisected at a point A(4, 3), which is nearer to $x$-axis, then its equation is :

The circumcentre of a triangle lies at the origin and its centroid is the mid point of the line segment joining the points $(a^2+1,a^2+1)$ and $(2a,-2a)$, $a\ne0$. Then for any a, the orthocentre of this triangle lies on the line :

For $a>b>c>0$, the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less than $2\sqrt2$. Then

Let $O(0,0)$, $P(3,4)$, $Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are

Let $ABCD$ be a quadrilateral with area 18, with side $AB$ parallel to the side $CD$ and $AB=2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is