Telangana Board Solutions for Chapter: The Straight Line, Exercise 2: Exercise 3(b)

If the sum of the reciprocals of the intercepts made by a variable straight line on the axes of coordinates is a constant, then prove that the line always passes through a fixed point.

Line $L$ has intercepts $a$ and $b$ on the axes of coordinates. When the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$ on the transformed axes. Prove that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}+\frac{1}{q^2}$.

Transform the equation $\frac{x}{a}+\frac{y}{b}=1$ into the normal form when $a>0$ and $b>0$. If the perpendicular distance of the straight line from the origin is $p$, deduce that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$.

A straight line passing through $A(-2,1)$ makes an angle of $30°$ with $\overleftrightarrow{OX}$ in the positive direction. Find the points on the straight line whose distance from $A$ is $4$ units.

Find the points on the line $3x-4y-1=0$ which are at a distance of $5$ units from the point $(3,2)$.

A straight line whose inclination with the positive direction of the $X-$axis, measured in the anti-clockwise sense is $\frac{\pi }{3}$ makes positive intercept on the $Y-$axis. If the straight line is at a distance of $4$ from the origin, find its equation.

A straight line $L$ is drawn through the point $A(2,1)$ such that its point of intersection with the straight line $x+y=9$ is at a distance of $3\sqrt{2}$ from $A$. Find the angle which the line $L$ makes with the positive direction of the $X-$axis.

A straight line $L$ with negative slope passes through the point $(8,2)$ and cuts positive coordinate axes at the points $P$ and $Q$. Find the minimum value of $OP+OQ$ as $L$ varies, where $O$ is the origin.