Animated Solution for Mathematics - Straight Lines: Suppose that the points (h,k), (1,2) and (-3,4) lie on the line L1. If a line L2 passing through the points (h,k) and (4,3) is perpendicular to L1, then k/h equals :

Imagine you are standing on a vast coordinate plane. You see two lines, $L_1$ and $L_2$, stretching out toward infinity. They are governed by strict geometric laws, and our mission is to find the point $(h,k)$ where these two paths cross.

This is a detective story where we use the clues of slope and perpendicularity to uncover a hidden coordinate.

We are told that three points—$(h,k)$, $(1,2)$, and $(-3,4)$—all reside on the same line $L_1$. By focusing on the two known points, $(1,2)$ and $(-3,4)$, we calculate the slope $m_1$ using the fundamental definition of change in $y$ over change in $x$:

With this slope, we use the point-slope form, $y-y_1=m_1(x-x_1)$, to define the identity of $L_1$. Substituting our point $(1,2)$, we get $y-2=-\frac{1}{2}(x-1)$.

After algebraic simplification, we arrive at the standard form: $x+2y=5$. This equation is the DNA of $L_1$; every point on that line, including our mystery point $(h,k)$, must satisfy this relationship.

Now, consider $L_2$. We know it passes through $(4,3)$, but its slope is defined by its relationship to $L_1$. Since they are perpendicular, the product of their slopes must be $-1$.

Given $m_1=-\frac{1}{2}$, we solve for $m_2$:

With a slope of $2$ and a point $(4,3)$, we construct the equation for $L_2$ using $y-3=2(x-4)$. Simplifying this, we find $2x-y=5$.

We are looking for the point $(h,k)$ that satisfies both $x+2y=5$ and $2x-y=5$. By rearranging the second equation to $y=2x-5$ and substituting it into the first, we solve for $x$:

Thus, $h=3$. Substituting this back into our expression for $y$, we find $k=2(3)-5=1$.

The mystery point is $(3,1)$. The final step is to find the ratio $\frac{k}{h}$, which results in $\frac{1}{3}$.