Question:
The ratio of the largest and shortest distances from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is
The ratio of the largest and shortest distances from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is
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Always reduce general circle form and use center-radius geometry for external point distances.
Updated On: May 19, 2025
- $15 : 13$
- $7 : 1$
- $3 : 2$
- $14 : 1$
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The Correct Option is D
Solution and Explanation
Convert circle to standard form:
$x^2 + y^2 - 14x - 10y = 151$
Complete the square:
$(x - 7)^2 + (y - 5)^2 = 225 \Rightarrow$ radius = $15$, center = $(7, 5)$
Distance from $(2, -7)$ to center = $\sqrt{(7 - 2)^2 + (5 + 7)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
Largest distance = $r + d = 28$, shortest = $r - d = 2$
Ratio = $28 : 2 = 14 : 1$
$x^2 + y^2 - 14x - 10y = 151$
Complete the square:
$(x - 7)^2 + (y - 5)^2 = 225 \Rightarrow$ radius = $15$, center = $(7, 5)$
Distance from $(2, -7)$ to center = $\sqrt{(7 - 2)^2 + (5 + 7)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
Largest distance = $r + d = 28$, shortest = $r - d = 2$
Ratio = $28 : 2 = 14 : 1$
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