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JEE Main 2018

JEE Main 2018 Mathematics question on Conic Sections

Tangents are drawn to the hyperbola 4x2−y2=364x^2 - y^2 = 36 at the points PP and QQ. If these tangents intersect at the point T(0,3)T(0, 3), then the area (in sq units) of △PTQ\triangle PTQ is
  1. A.
    (a) 45545\sqrt{5}
  2. B.
    (b) 54354\sqrt{3}
  3. C.
    (c) 60360\sqrt{3}
  4. D.
    (d) 36536\sqrt{5}
Show answer and solution

Correct option: A

Solution

The hyperbola is 4x² - y² = 36, or x²/9 - y²/36 = 1. T(0,3) is the pole, so the chord of contact PQ has equation: 4(0)x - (3)y = 36, giving y = -12.

Find P and Q by substituting y = -12 into the hyperbola: 4x² - 144 = 36, so 4x² = 180, x² = 45, x = ±3√5. Thus P = (3√5, -12) and Q = (-3√5, -12).

Length PQ = 6√5. The height from T(0,3) to line y = -12 is |3-(-12)| = 15.

Area = (1/2) × 6√5 × 15 = 45√5.

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