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
Show Hint
For a point at distance \(d\) from the center of a circle of radius \(r\), the maximum and minimum distances to the circle are \(r+d\) and \(|r-d|\) respectively.
Updated On: Jun 22, 2026
- \(15:13\)
- \(7:1\)
- \(3:2\)
- \(14:1\)
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The Correct Option is D
Solution and Explanation
Step 1: Convert the circle into standard form.
Given equation of the circle is
\[ x^2+y^2-14x-10y-151=0 \] Group the \(x\) and \(y\) terms together:
\[ (x^2-14x)+(y^2-10y)=151 \] Complete the squares:
\[ (x^2-14x+49)+(y^2-10y+25)=151+49+25 \] \[ (x-7)^2+(y-5)^2=225 \] Thus, the center and radius are
\[ C=(7,5), \qquad r=15 \]
Step 2: Find the distance of the given point from the center.
The given point is
\[ P=(2,-7) \] Distance between \(P\) and the center \(C(7,5)\) is
\[ PC=\sqrt{(7-2)^2+(5-(-7))^2} \] \[ =\sqrt{5^2+12^2} \] \[ =\sqrt{25+144} \] \[ =\sqrt{169} \] \[ =13 \]
Step 3: Find the largest and shortest distances.
For a point inside a circle:
\[ \text{Largest distance}=r+d \] and
\[ \text{Shortest distance}=r-d \] where \(d\) is the distance from the center.
Here,
\[ r=15,\qquad d=13 \] Therefore,
\[ \text{Largest distance}=15+13=28 \] and
\[ \text{Shortest distance}=15-13=2 \]
Step 4: Find the required ratio.
\[ \text{Ratio}=\frac{28}{2} \] \[ =14:1 \]
Step 5: Final conclusion.
Hence, the required ratio is
\[ \boxed{14:1} \]
Given equation of the circle is
\[ x^2+y^2-14x-10y-151=0 \] Group the \(x\) and \(y\) terms together:
\[ (x^2-14x)+(y^2-10y)=151 \] Complete the squares:
\[ (x^2-14x+49)+(y^2-10y+25)=151+49+25 \] \[ (x-7)^2+(y-5)^2=225 \] Thus, the center and radius are
\[ C=(7,5), \qquad r=15 \]
Step 2: Find the distance of the given point from the center.
The given point is
\[ P=(2,-7) \] Distance between \(P\) and the center \(C(7,5)\) is
\[ PC=\sqrt{(7-2)^2+(5-(-7))^2} \] \[ =\sqrt{5^2+12^2} \] \[ =\sqrt{25+144} \] \[ =\sqrt{169} \] \[ =13 \]
Step 3: Find the largest and shortest distances.
For a point inside a circle:
\[ \text{Largest distance}=r+d \] and
\[ \text{Shortest distance}=r-d \] where \(d\) is the distance from the center.
Here,
\[ r=15,\qquad d=13 \] Therefore,
\[ \text{Largest distance}=15+13=28 \] and
\[ \text{Shortest distance}=15-13=2 \]
Step 4: Find the required ratio.
\[ \text{Ratio}=\frac{28}{2} \] \[ =14:1 \]
Step 5: Final conclusion.
Hence, the required ratio is
\[ \boxed{14:1} \]
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