How to efficiently remove a query string by Key from a Url?

This works well:

public static string RemoveQueryStringByKey(string url, string key)
{                   
    var uri = new Uri(url);

    // this gets all the query string key value pairs as a collection
    var newQueryString = HttpUtility.ParseQueryString(uri.Query);

    // this removes the key if exists
    newQueryString.Remove(key);

    // this gets the page path from root without QueryString
    string pagePathWithoutQueryString = uri.GetLeftPart(UriPartial.Path);

    return newQueryString.Count > 0
        ? String.Format("{0}?{1}", pagePathWithoutQueryString, newQueryString)
        : pagePathWithoutQueryString;
}

an example:

RemoveQueryStringByKey("https://www.google.co.uk/search?#hl=en&output=search&sclient=psy-ab&q=cookie", "q");

and returns:

https://www.google.co.uk/search?#hl=en&output=search&sclient=psy-ab