Multiple choice

The tangents drawn from the origin to the circle x2 + y2 – 14x + 2y + 25 = 0 are

  1. Parallel

  2. Perpendicular

  3. Coincident

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The circle is x^2 + y^2 - 14x + 2y + 25 = 0. The origin (0,0) is outside the circle (0+0-0+0+25 > 0). Tangents from a point to a circle are perpendicular if the point lies on the director circle. Director circle is x^2 + y^2 - 14x + 2y + 25 - r^2 = 0. r^2 = g^2 + f^2 - c = 49 + 1 - 25 = 25. Director circle: x^2 + y^2 - 14x + 2y = 0. Since (0,0) satisfies this, the tangents are perpendicular.

AI explanation

Rewriting the circle equation x2 + y2 – 14x + 2y + 25 = 0 in standard form gives (x - 7)2 + (y + 1)2 = 25. This shows the circle has its center at (7, -1) and a radius of 5. The distance from the origin (0,0) to the center (7, -1) is the square root of (7 squared plus 1 squared), which equals the square root of 50. Since the distance from the origin to the center equals the square root of the quantity (radius squared plus radius squared), the pair of tangents drawn from the origin to the circle are perpendicular to each other.

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