Circles
Circle
nta_abhyas_2025
Grade None
Question:
If the lengths of tangents from $P(1,3)$ and $Q(3,7)$ to a circle arc $\sqrt{2}$ units and $\sqrt{18}$ units respectively, then the length of the tangent from $R(7,15)$ to the same circle is
√98 units
√170 units
√50 units
None of these
Step-by-Step Solution
Key Concept: The tangent length from an external point to a circle equals $\sqrt{x_0^2 + y_0^2 + 2px_0 + 2fy_0 + C}$ where the circle equation is $x^2 + y^2 + 2px + 2fy + C = 0$.
Let the circle be $x^2 + y^2 + 2px + 2fy + C = 0$. The length of tangent from point $(7, 15)$ to this circle is 1 unit. This gives: $49 + 225 + 14p + 30f + C = 1$, so $14p + 30f + C = -273$ (equation 1). Additionally, $9 + 49 + 6p + 14f + C = 18$ (equation 2) from another condition. Solving these equations yields $24m^2 - 14m - 24 = 0$, giving product $m_1 m_2 = -1$, confirming perpendicularity. The answer is $\sqrt{170}$ units.
Correct Answer: 170