Circles Question - None

Question

Given the equation of the circle is $x^2 + y^2 + 4x - 10y - 7 = 0$, find the minimum distance from the point $(4, -3)$ to any point on the circle.

Accepted Answer

Step 1: Convert the equation to standard form to find centre and radius.

Step 2: Centre is $(-2, 5)$ and radius $r = \sqrt{4 + 25 + 7} = \sqrt{36} = 6$

Step 3: Find distance from point $(4, -3)$ to centre $(-2, 5)$.

Step 4: $d = \sqrt{(-2 - 4)^2 + (5 + 3)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Step 5: Verify the point lies outside the circle: $(4)^2 + (-3)^2 + 4(4) - 10(-3) - 7 = 64 > 0$

Step 6: Minimum distance = distance to centre - radius = $10 - 6 = 4$

∴ Answer is $4$.

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