Custom search query by category id

Try this code

<?php
$args = array(
     's' => $_REQUEST[ 'search' ],
     'cat' => 'category_id',
     'paged' => $paged
);
// The Query
$the_query = new WP_Query( $args );

// The Loop
while ( $the_query->have_posts() ) : $the_query->the_post();
    //inside the loop
    the_title();
    the_content();
endwhile;

// Reset Post Data
wp_reset_postdata();

?>

Hope this help