What are the notable contributions of Newton and Leibniz in analytic geometry?

Answer 1

Newton is generally credited with the generalized binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula), and was the first to use power series with confidence and to revert power series. He also discovered a new formula for calculating pi.

In 1676 Leibniz conceived a kind of "algebra of thought," modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation) discussed in 3.2, a universal concept language, and more.

Answer 2

They're generally regarded as coinventors of calculus.

If you bracket them together, that's the point.

I'd say Newton did more, actually, e.g. by pointing out what Kepler's Laws really meant in mathematical terms.