Straight Line and its Equations

If the point $\left(\alpha , \frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta +y\sin \theta =7, \theta \in \left(0, \frac{\mathrm{\pi }}{2}\right)$ between the co-ordinates axes, then $\alpha$ is equal to

If the base of an isosceles triangle is of length $2a$ and length of altitude dropped to the base is $h$, then the distance from the mid-point of the base to the side of the triangle is

If a point $P\left(x,y\right)$ from the line drawn cuts coordinate axes at $A$ and $B$( with $A$ on $x-$axis and $B$ on $y-$axis) satisfies $\frac{\alpha x^2}{P{B}^2}+\frac{{\displaystyle \beta y^2}}{{\displaystyle P{A}^2}}=1$, then $\alpha +\beta$ is

If $\frac{ax}{\sqrt{a^2+b^2}}+\frac{by}{\sqrt{a^2+b^2}}=-\frac{c}{\sqrt{a^2+b^2}}$ is perpendicular form of straight line, then

The normal form of the line $x+y\sqrt{3}+7=0$ is

Find the gradient of the line:

passing though $(0,2.5)$, and $\left(\frac{-1}{3},5\right)$ 

Find the gradient (slope) of the line passing through the following pair of points:

$\mathrm{A}(-1.3,2.2)$ and $B\left(1\frac{1}{5},-0.3\right).$

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively. If the area of the triangle $ABC$ is $5$ sq.units, then the equation of the side $BC$ is

A straight line having slope $\frac{1}{\mathrm{\pi }}$ passes through ordered pairs $\left(\lambda ,\mu \right)$ satisfying the equation ${\cos }^2\mu +4\lambda -{\lambda }^2=5, \mu \in \left[0,\mathrm{\pi }\right]$. The equation of line can be

The angle included between the straight lines $x=0$ and $y=\sqrt{3}x+5$ is _________

The gradient of the straight line which passes through the point $(-3, 6)$ and through the middle point of the line segment joining the points $(4, -5)$ and $(-2, 9)$ is _____

The equation of the straight line which passes through the point $(3, 4)$ and cuts the parts from the coordinate axes, equal in both magnitude and sign is ______

The equation of the straight line which passes through $(0, -4)$ and is inclined to the $x-$axis at an angle $45°$ is ______

The angle made by the straight line $\sqrt{3}x+y=1$ with the negative direction of $x-$axis is ______

If the gradient of the straight line joining the points $(3k, 4)$ and $(4, -3k)$ is $-3,$ then $k=$

The $y-$axis divides the line segment joining the points $(-3, -4)$ and $(1, -2)$ internally in the ratio----

If the coordinates of the centroid and two vertices of a triangle are respectively $(2, 7), (4, 8), (-2, 6),$ then the coordinates of the third vertex are _________