making an archive filter with a drop down, based on two categories

Hello,

I’m trying to make a drop down menu that acts like a filter for an archive. Basically I would like it, when selected to show all the posts in the option selected in the drop down + the cat ID I’m pulling before the drop down. Here’s what I have so far, I just don’t know how to work $cat into the bottom…

<?php
// get the first specific category to filter by
$cat=get_cat_ID('Teacher\'s Lounge');
echo $cat;
?>

<?php
//get the second category to filter by
$args = array(
'child_of'=> 44,
'depth' => 1,
)

?>

<?php wp_dropdown_categories($args); ?>
<script type="text/javascript">
  var dropdown = document.getElementById("cat");
  function onCatChange() {
	if ( dropdown.options[dropdown.selectedIndex].value > 0 ) {
		location.href =
		"<?php echo get_option('home');?>/?cat="+dropdown.options[dropdown.selectedIndex].value;
	}
    }
    dropdown.onchange = onCatChange;
</script>

My thought is that it’s something along the lines of:

<?php echo get_option('home');?>/?cat="&&$cat+dropdown.options[dropdown.selectedIndex].value;

but I am not sure of syntax.

suggestions?

Thank you.

Nadine.