A circle has centre at $(3, - 2)$ and passes through the point $(8, 1)$ . Find the equation of the circle and then determine the length of a tangent drawn from the external point $(5, 5)$ to this circle.

hard 5 marks Class 9 · Mathematics · Coordinate Geometry

Answer

(x - 3)^{2} + (y + 2)^{2} = 34,; tangent length = 19

The correct answer is $(x - 3)^{2} + (y + 2)^{2} = 34,; tangent length = 19$ — Hence equation is $(x - 3)^{2} + (y + 2)^{2} = 34$ and tangent length is $19$ units.

Step-by-step solution

Find radius: distance from centre $(3, - 2)$ to point $(8, 1)$ : $r = (8 - 3)^{2} + (1 + 2)^{2} = 25 + 9 = 34$ .
Write circle equation: $(x - 3)^{2} + (y + 2)^{2} = r^{2} = 34$ .
Compute distance $d$ from external point $(5, 5)$ to centre: $d = (5 - 3)^{2} + (5 + 2)^{2} = 4 + 49 = 53$ .
Length of tangent $l$ is $d^{2} - r^{2} = 53 - 34 = 19$ .
Hence equation is $(x - 3)^{2} + (y + 2)^{2} = 34$ and tangent length is $19$ units.

Common mistake: using

d^{2} - r

instead of

d^{2} - r^{2}

in the tangent length formula

Concept tested: circle equation and tangent length

Published 13 October 2026 · How this solution was prepared and checked

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A circle has centre at (3,−2) and passes through the point (8,1). Find the equation of the circle and then determine the length of a tangent drawn from the external point (5,5) to this circle.

Step-by-step solution

A circle has centre at $(3, - 2)$ and passes through the point $(8, 1)$ . Find the equation of the circle and then determine the length of a tangent drawn from the external point $(5, 5)$ to this circle.