wp_dropdown_categories() searching is not working with CPT, please review my code

Question

Searching is not working with CPT with custom taxonomy

<form id="searchform" method="get" action="<?php bloginfo('url'); ?>">
        <input type="text" name="s" id="s" size="15" />
        <?php //wp_dropdown_categories('show_option_none=Select category&taxonomy=type'); ?>
        <?php wp_dropdown_categories('depth=0&orderby=name&hide_empty=1&show_option_all=Search Everything&name=course&taxonomy=type'); ?>
        <input type="submit" value="Search" />
    </form>

But with default post and category is working with this code

<form id="searchform" method="get" action="<?php bloginfo('url'); ?>">
    <input type="text" name="s" id="s" size="15" />
    <?php //wp_dropdown_categories('show_option_none=Select category&taxonomy=type'); ?>
    <?php wp_dropdown_categories('depth=0&orderby=name&hide_empty=1&show_option_all=Search Everything'); ?>
    <input type="submit" value="Search" />

Answer 1

Working drop down in search for CPT

<?php

  $categories = get_categories('taxonomy=custom-tax');

  $select = "<select name='cat' id='cat' class='postform'>n";
  $select.= "<option value='-1'>Select category</option>n";

  foreach($categories as $category){
    if($category->count > 0){
        $select.= "<option value='".$category->slug."'>".$category->name."</option>";
    }
  }

  $select.= "</select>";

  echo $select;
?>

<script type="text/javascript"><!--
    var dropdown = document.getElementById("cat");
    function onCatChange() {
        if ( dropdown.options[dropdown.selectedIndex].value != -1 ) {
            location.href = "<?php echo home_url();?>/category/"+dropdown.options[dropdown.selectedIndex].value+"/";
        }
    }
    dropdown.onchange = onCatChange;
--></script>