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Let P(r, s) with r < 0 be a variable point on the ellipse C1: x²/16 + y²/9 = 1. From P, two tangents PA and PB are drawn to the parabola C2: y² - 4x = 0, with A and B the points of contact. Which of the following statements is correct?

1. the midpoint of chord AB lies on the curve x²/9 + (y² - 2x)²/64 = 1
2. the midpoint of chord AB lies on the curve y²/9 + (y² - 2x)²/64 = 1
3. the number of common tangents of curves C1 and C2 is 4
4. the minimum integral value of r + s is -5

Correct answer: the midpoint of chord AB lies on the curve x²/9 + (y² - 2x)²/64 = 1

Solution

Expressing the midpoint of the chord of contact and substituting P on the ellipse yields the locus x²/9 + (y² - 2x)²/64 = 1.

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