Yes, Jupiter has asteroids locked in orbit with it at all of its stable Lagrangian Points.

In lagrangian we use scalar quantity which is much easier to solve then vector quantites used in newtonian.also lagrangian is invariant under gauge transformation and the same not hold good for newtonian.

True. Points are geometric objects with no dimensions.

SOHO (Solar and Heliospheric Observatory) was launched into the Earth/Sun L1 Lagrangian point in 1995. This point balances the gravity from the Sun and Earth and allows for very little energy to remain in a stable orbit. There are 5 Lagrangian points for SOHO but L1 is the best positioned for Earth communications.

That happens relatively close to Earth, because the Sun has a much stronger gravitational field than Earth.

It's inside the Earth's orbit about 260,000 kilometers from Earth, in the direction of the Sun.

There's a similar place about the same distance from Earth, but outside the Earth's orbit.

The Earth is constantly moving relative to the Sun, so there is no fixed place in space where this happens.

Notice that these places are not the "Lagrangian points". They are a related, but different thing.

I've a feeling this question may indeed be about the Lagrangian points.

In that case the answer is: "at one of the 5 Lagrangian points".

However, the forces balance there only if you include the "centripetal force"

required to keep a small object, at one of these points, in a stable orbit.