Straight Line And Pair Of Straight Lines Ques 82

Solution:

82. Let the vertices of a triangle be, $O(0,0) A(a, 0)$ and $B(b, c)$ equation of altitude $B D$ is $x=b$.

Slope of $O B$ is $\frac{c}{b}$.

Slope of $A F$ is $-\frac{b}{c}$.

Now, the equation of the altitude from $A$ to $F$ is

$ y-0=-\frac{b}{c}(x-a) $

Suppose, $B D$ and $O E$ intersect at $P$.

Coordinates of $P$ are $[b, b \quad \frac{(a-b)}{c}]$

Let $m_1$ be the slope of $OP=\frac{a-b}{c}$

and $\quad m_2$ be the slope of $AB=\frac{c}{b-a}$

Now, $\quad m _1 m _2=(\frac{a-b}{c}) \quad (\frac{c}{b-a})=-1$

We get that the line through $O$ and $P$ is perpendicular to $AB$.