Let the tangent drawn to the parabola y²=24 x at the point (α, β) is perpendicular to the line 2 x+2 y=5. Then the normal to the hyperbola x²/α²-y²/β²=1 at the point (α+4, β+4) does NOT pass through the point :

Solution

Maths sir · 3 steps

Use the parabola to pin down α and β, write the hyperbola's normal, then test each point in it.

1. Find the point. The line 2x+2y=5 has slope -1, so the tangent has slope 1. For y²=24x, dy/dx=12/y, so 12/β=1 gives β=12. Then α=144/24=6.
2. Set up the hyperbola. The hyperbola is x²/36-y²/144=1, and the point is (10, 16). It lies on the curve: 100/36-256/144=(400-256)/144=1.

   a²x/x₁+b²y/y₁=a²+b² ⇒ 36x/10+144y/16=180 ⇒ 2x+5y=100

   Point	2x+5y	On the normal?
   (a) (25, 10)	50+50=100	yes
   (b) (20, 12)	40+60=100	yes
   (c) (30, 8)	60+40=100	yes
   (d) (15, 13)	30+65=95	no

3. So. Only (15, 13) fails to satisfy 2x+5y=100, so the normal does not pass through it. The answer is (d).

The idea to keep. Normal to x²/a²-y²/b²=1 at (x₁,y₁) is a²x/x₁+b²y/y₁=a²+b². Clear the fractions first, and checking points becomes one line each.

Answer from the NTA final answer key. Solution by PrepWiser.