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Area of Tangent Triangle on Parabola y²-2x=8y-20 | JEE

JEE Maths question with a full step-by-step solution.

Question
The area of the triangle formed by the tangents at the points (4,6)(4, 6), (10,8)(10, 8) and (2,4)(2, 4) on the parabola y2−2x=8y−20y^2 - 2x = 8y - 20 is (in sq. units):
A44
B22correct
C11
D88
Solution
Step 1: All three points satisfy y2−2x−8y+20=0y^2 - 2x - 8y + 20 = 0. For a parabola the tangent triangle has half the area of the triangle on the points themselves. Step 2: The triangle on (4,6),(10,8),(2,4)(4,6),(10,8),(2,4) has area
12∣4(8−4)+10(4−6)+2(6−8)∣=4⇒required=2.\frac{1}{2}\big|4(8 - 4) + 10(4 - 6) + 2(6 - 8)\big| = 4 \Rightarrow \text{required} = 2.
 Correct answer: (2)
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