Embibe Experts Solutions for Chapter: Point and Straight Line, Exercise 3: EXERCISE-3

If the point $P\left(3,5\right)$ divides the line joining $A\left(1,3\right)$ & $B\left(7,9\right)$ and harmonic conjugate of $P$ w.r.t.$A$ & $B$ is $Q\left(a,b\right)$, then $a+b$ is

If two vertices of a triangle are $\left(4,-3\right)$ & $\left(-2,5\right)$. If the orthocentre of the triangle is at $\left(1,2\right)$ then coordinates of the third vertex is $\left(p,q\right)$, then $\left(\frac{p+q}{4}\right)$ is equal to

The line $3x+2y=24$ meets the $y$-axis at $A$ & the $x$-axis at $B$. The perpendicular bisector of $AB$ meets the line through $\left(0,-1\right)$ parallel to $x$-axis at $C$. If the area of the triangle $ABC$ is $\lambda$ sq. units, then value of $\lambda$ is equal to

If a line $ax+by+c=0$ is such that its segment between the straight lines $5x-y+4=0$ and $3x+4y-4=0$ is bisected at the point $\left(1,5\right)$, then $\left(a-b+c\right)$ is

If the straight line drawn through the point $P\left(\sqrt{3}, 2\right)$ and inclined at an angle $\frac{\pi }{6}$ with the $x$ - axis meets the line $\sqrt{3}x-4y+8=0$ at $Q,$ then the length of $PQ$ is

The points $\left(1,3\right)$ and $\left(5,1\right)$ are two opposite vertices of a rectangle. The other two vertices lie on the line $y=2x+c$, then $c$ is -

Two sides of a rhombus $ABCD$ are parallel to the lines $y=x+2$ and $y=7x+3$. If the diagonals of the rhombus intersect at the point $\left(1,2\right)$ and the vertex $A$ is on the $y$-axis and possible co-ordinates of $A$ are $\left(0,a\right)$ and $\left(0,b\right)$ then $4\left(a+b\right)$ is equal to

In a triangle $ABC$, $D$ is a point on $BC$ such that $\frac{BD}{DC}=\frac{AB}{AC}$. The equation of the line $AD$ is $2x+3y+4=0$ & the equation of the line $AB$ is $3x+2y+1=0$. If the equation of the line $AC$ is $ax+by+c=0$, then $\left(\frac{a+b+c}{10}\right)$ is equal to