Answer:
see explanation
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (h, 0) and (x₂, y₂ ) = (0, k)
m = [tex]\frac{k-0}{0-h}[/tex] = - [tex]\frac{k}{h}[/tex]
note the line crosses the y- axis at (0, k) ⇒ c = k
y = - [tex]\frac{k}{h}[/tex] x + k ← equation of line
Given that (3, 3) lies on the line then
3 = - [tex]\frac{3k}{h}[/tex] + k ( add [tex]\frac{3k}{h}[/tex] to both sides )
3 + [tex]\frac{3k}{h}[/tex] = k ( multiply through by h to clear the fraction )
3h + 3k = hk ← factor out 3 on the left side
3(h + k) = hk ← divide both sides by hk )
[tex]\frac{3(h+k)}{hk}[/tex] = 1 ( divide both sides by 3 )
[tex]\frac{h+k}{hk}[/tex] = [tex]\frac{1}{3}[/tex], that is
[tex]\frac{h}{hk}[/tex] +[tex]\frac{k}{hk}[/tex] = [tex]\frac{1}{3}[/tex]
[tex]\frac{1}{k}[/tex] + [tex]\frac{1}{h}[/tex] = [tex]\frac{1}{3}[/tex] ← as required
