YS Rajasekhara Reddy, the chief minister for Andhra Pradesh state, of which Hyderabad is the main city, called the bombings "a terrorist act" and urged people to remain calm.
Hyderabad's city administrator said that police had some "already have some leads", without elaborating.
One blast took place at the Lumbini amusement park where a laser show was taking place.
"One terrorist group or the other, which is bent on destroying the unity of the country, is certainly involved in the blasts in Hyderabad"
Sriprakash Jaiswal, federal minister of state for home
Reuters reported a police officer as saying that seven people were killed immediately in the park explosion, and two died later in hospital.
Reddy said about 500 people were in the park's auditorium at the time of the explosion.
Dismembered limbs, clothes and shoes of people were scattered on the ground at the scene of the attack.
"People were running away, some who were hurt had blood streaming from their bodies," a security guard said.
Injured people were rushed to hospital in three-wheel scooters and cars.
A police officer told Reuters that another explosion at a street-side restaurant was caused by a exploding gas cylinder and killed 11 people.
A total of 50 people were wounded in the two explosions.
NDTV showed what appeared to be at least two bodies slumped under rows of blue plastic seating in the park, both with blood-soaked clothing.
Hyderabad, a city of seven million people, about 40 per cent of them Muslim, has long dealt with Hindu-Muslim tensions and occasional violence.
Sriprakash Jaiswal, federal minister of state for home, said the explosions were part of an effort to undermine the city's mixed Hindu-Muslim community.
"One terrorist group or the other, which is bent on destroying the unity of the country, is certainly involved in the blasts in Hyderabad," he said.
The attacks come three months after 11 people died in a bomb blast at the city's Mecca mosque, which police have yet to name suspects for.
After the mosque attack, five people died in clashes between security forces and Muslim protesters angered by what they said was a lack of police protection.
Other major cities in India have been targeted by bomb attacks.
In July 2006, seven bombs detonated on Mumbai's rail network, killing 186 people.
"We're seeing a pattern of attacks every two to three months somewhere or other in the country on soft targets," Ajai Sahni, head of the New Delhi-based Institute for Conflict Management, said.
"They appear to be linked to what we call Pakistan-backed Islamic terrorist groups."
India has accused Islamic fighters in Indian Kashmir of being behind several attacks.
India and Pakistan both claim the divided region in full.