Boom. The vertical line x=0 (the y-axis). Does it touch y²=4x? Only at (0,0). Yes, it’s a tangent. He had missed it because his formula had a blind spot.
He stared for an hour. Then, he re-read the theory in Cengage, not the solved problems. Buried in a small, often-ignored "Note" box was a line: "For a parabola y² = 4ax, from an external point, you may have 0, 1, or 2 real tangents. Always check the discriminant method. Never rely solely on the slope form; it fails for vertical lines." maths cengage jee mains
Arjun pointed at the coaching modules. "They are too easy. The exam is different." Only at (0,0)
He tried. He failed. He drew graphs. He failed again. He derived the equation of a tangent in slope form (y = mx + 1/m). He plugged in (0, -2). He got -2 = 0 + 1/m → m = -1/2. One tangent. But his gut screamed there was another—a vertical one. He stared for an hour
"This," he said. "But only if you are ready to bleed. The book doesn't give you marks. The process of wrestling with it does. Every star you chase, every wrong answer you autopsy, every time you choose the hard problem over the easy one—that’s not just studying. That’s building a mind that JEE Main cannot trick."
Find the number of distinct real tangents that can be drawn from the point (0, -2) to the curve y² = 4x.
That night, Arjun turned to a random page in by G. Tewani. Chapter: Application of Derivatives . Topic: Tangents and Normals . He chose a problem marked with the infamous red star (*) – the "advanced" level.