Miscellaneous Exercise on Chapter 10

Find the value of $k$ for which the line $\left(k-3\right)x-\left(4-k^2\right)y+k^2-7k+6=0$ is parallel to the $x$-axis.

Find the value of $p$ so that the three lines $3x+y-2=0,px+2y-3=0$ and $2x-y-3=0$ may intersect at one point.

The hypotenuse of a right-angled triangle has its ends at the points $\left(1,3\right)$ and $\left(-4,1\right)$. Find the equation of the legs (perpendicular sides) of the triangle.

If sum of the perpendicular distances of a variable point $P\left(x,y\right)$ from the lines $x+y-5=0$ and $3x-2y+7=0$ is always $10.$ Show that $P$ must move on a line.

Find an equation of the line which is equidistant from parallel lines $9x+6y-7=0$ and $3x+2y+6=0.$

A ray of light passing through the point $\left(1,2\right)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $\left(5,3\right).$ Find the coordinates of $A.$

Prove that the product of the lengths of the perpendiculars drawn from the points $\left(\sqrt{a^2-b^2},0\right)$ and $\left(-\sqrt{a^2-b^2},0\right)$ to the line $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ is $b^2.$

A person standing at the junction (crossing) of two straight paths represented by the equations $2x-3y+4=0$ and $3x+4y-5=0$ wants to reach the path whose equation is $6x-7y+8=0$ in the least time. Find equation of the path that he should follow.