The line $y = 10$ and $3y + 4x = 32$ are tangent to a circle $C$ at the points $(-2,10)$ and

Question

The line $y = 10$ and $3y + 4x = 32$ are tangent to a circle $C$ at the points $(-2,10)$ and $(2,8)$ respectively. Show that the equation of $C$ is $(x + 2)^2 + (y - 5)^2 = 25$.[5] Explain if the $x$-axis is tangent to $C$.[2] Write down the equations of the two vertical tangents.[2] Two points $P$ and $Q$ lie on the circle and the length of $PQ$ is 4 units. Calculate the shortest distance from the centre of the circle to the line $PQ$.[2]

Accepted Answer

(a) $(x+2)^2+(y-5)^2=25$ (shown). (b) Yes, tangent at $(-2,0)$. (c) $x=-7$ and $x=3$. (d) $\sqrt{21}\approx4.58$ units.

2024 GESS PRELIMS P2 Q3

2024 GESS PRELIMS P2 Q3

Solution:

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