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Some homeowners plant elaborate gardens. Others splurge on tennis courts or swimming pools. Alan Wilzig prefers something a bit racier.

In the back yard of his 150-year-old Dutch Colonial-style home in the Hudson Valley town of Taghkanic, he has built a 1.15-mile-long, 40-foot-wide, bidirectional racetrack — heavy on the hairpin turns and smooth as a billiard table — designed for high-performance motorcycles and cars.

“You’re going fast enough to make your ass pucker,” Wilzig, 52, told The Post. “The f—ing hairs on your neck stand up.”

The lobster-claw-shaped course is the only personal-use, professional-quality private racetrack in the world, according to Wilzig. It boasts nine turns, 80 feet of elevation changes, grass-covered boundaries and FoamAir fences outside the corners to soften the impact of going off course. (There have been four accidents on the track but no injuries.) Wilzig, who once made headlines for having co-owned a castle in Water Mill, LI, with his brother Ivan, describes the track as a “field of dreams for motor sports.”

The impetus for it came in the late 1990s after a motorcycle-riding buddy in the Hamptons nearly got killed by an automobile; the idea of street-riding became increasingly unappealing to Wilzig. “After 100,000 miles of around-the-world motorcycle riding without incident,” he said, “I realized it was only a matter of time before somebody pulled out in a fancy car and hit me.”

Wilzig financed his property purchase with cash from the sale of the Trust Company of New Jersey bank in 2004. Long controlled by Wilzig’s father, Siggi B., who died in 2003, it went for $726 million. In 2005, Alan took his cut of the profits and bought the 275-acre spread upstate for $3.35 million.

“I bought this property with the intention of building a racetrack,” said Wilzig, who served as the bank’s CEO and now focuses primarily on philanthropy. “Zoning here allowed for it — unlike in the Hamptons.”

Before Wilzig began designing his track, however, he knew one thing for sure: He would not be driving German cars and bikes on it. His father was an Auschwitz survivor, and 59 of his relatives perished in Nazi death camps. His speedy collection includes 110 motorcycles and a dozen cars, but no BMWs, Porsches or Mercedes. “I don’t need to see that three-pointed star [the Mercedes-Benz hood ornament] when I know it was the last thing my grandparents saw before they were murdered by the Nazis,” he said. (Daimler-Benz is known to have used concentration camp detainees as slave labor to manufacture cars.)

In 2006, news of the proposed track broke in The Post, and town residents freaked out, fearing noise and commercial usage. When construction of the track began in 2007, locals halted completion via a court injunction. Wilzig had to spend $500,000 in legal fees and two years in court to continue construction. The course was finished in May 2010 at a total cost of around $3 million.

Although Wilzig built the track as a place for zipping around curves with motor-savvy friends — including celebrity hairdresser Oscar Blandi and film and TV director Ben Younger (who visited earlier this month and holds the motorcycle track record: a lap in 52.8 seconds) — the recently divorced father of two has one riding partner whom he prefers above all others: his 23-year-old girlfriend of nearly three years, Clemence Lapeyre.

Lapeyre, who graduated from Duke University with a degree in computer science last year, is as passionate about racing as Wilzig is. “The most fun I ever had here was this past summer when we raced motorcycles on the track every day,” said Wilzig, who figures he’s done some 15,000 laps around his track. “I would wake up with my girlfriend at 7 in the morning, watch a live Formula One race from Singapore or Bahrain or wherever on TV, then go downstairs and make breakfast for my 10- and 11-year-old children — who, sadly, do not share my enthusiasm for racing. After that, Clem and I would spend two hours chasing each other around the track before coming back, making love for a half-hour and then enjoying three or four hours in the pool with the kids. What could be a better day?”

One recent guest of a guest was New York-based art consultant Chessa Ferro. The visit was her first, and she said that even whizzing along the track in the passenger seat, hitting triple-digit speeds, was not for the faint of heart. “The forces against your body when you go that fast are pretty intense,” she said, adding that she did not feel up to the task of getting behind the wheel herself, but nevertheless relished her high-speed thrills. “We spun out of control three times and wound up on the grass — there’s nothing to hit. I was cracking up. Everybody thought I was such a weirdo. But I thought it was hilarious.”

Making things all the more over-the-top is Wilzig’s “moto museum,” a three-story steel-and-glass structure where his $5 million collection of cars and bikes — including Ducatis, Suzukis and Bimotas — is on display.

There’s the Lola, which is the model car that everybody recognizes from the “Speed Racer” cartoons; a motorcycle that once belonged to entrepreneur Malcolm Forbes (“It had its own suitcase so that he could easily transport it on his 727,” Wilzig said); and a 2015 McLaren, which goes for $350,000, achieves speeds of 205 mph and comes loaded with a battery of high-tech features.

“[That] car makes an average driver into a good driver and a good driver into [four-time Formula One champion] Lewis Hamilton,” Wilzig said, pointing out an auxiliary brake situated in one of the wheels and attached to the car’s computer. “If I’m making a turn too fast, the computer senses it and the brake caliper gets pinched. That rotates the car around turns better than you could on your own.”

Impressive as Wilzig’s collection is — complete with 100 helmets and racing suits designed to fit everyone from a 6-year-old kid to a 6-foot-4, 240-pound man — he acknowledges that it pales in comparison to those of deeper-pocketed car lovers. But Wilzig is proud to have something that they do not.

“A guy buys a Ferrari driven by Steve McQueen for $27 million and nine times out of 10, he takes it out once a week so he can drive to the coffee place in Greenwich [Conn.],” said Wilzig. “Come here and you can actually drive the cars in the way they were meant to be driven. I can’t think of anything more fun.”