SQL Azure: Drop Constraint and Index

This may be a stupid question, but I'm not a DBA and just wanna make sure of what I'm doing before I make a mistake.

Here is my question: When dropping Constraints, are their indexes dropped as well?

Basically, I'm asking because I want to modify an existing constraint. I want to change the foreign key to another column.

Here was the original query to create them:

-- Creating foreign key on [Industry_Id] in table 'Courses'
ALTER TABLE [dbo].[Courses]
ADD CONSTRAINT [FK_CourseIndustry]
    FOREIGN KEY ([Industry_Id])
    REFERENCES [dbo].[Industries]
        ([Id])
    ON DELETE NO ACTION ON UPDATE NO ACTION;

-- Creating non-clustered index for FOREIGN KEY 'FK_CourseIndustry'
CREATE INDEX [IX_FK_CourseIndustry]
ON [dbo].[Courses]
    ([Industry_Id]);
GO

Here is what I'm thinking of doing:

IF OBJECT_ID(N'[dbo].[FK_CourseIndustry]', 'F') IS NOT NULL
    ALTER TABLE [dbo].[Courses] DROP CONSTRAINT [FK_CourseIndustry];
GO

-- Creating foreign key on [IndustryId] in table 'Courses'
ALTER TABLE [dbo].[Courses]
ADD CONSTRAINT [FK_CourseIndustry]
    FOREIGN KEY ([IndustryId])
    REFERENCES [dbo].[Industries]
        ([Id])
    ON DELETE NO ACTION ON UPDATE NO ACTION;

-- Creating non-clustered index for FOREIGN KEY 'FK_CourseIndustry'
CREATE INDEX [IX_FK_CourseIndustry]
ON [dbo].[Courses]
    ([IndustryId]);
GO

Is that enough and will it work? Honestly, I don't mind if it's not an elegant solution just so long as it works.

Any piece of advise or information would be highly appreciated. Thanks!