$A(1, 7)$ and $B(9, 7)$ lie on a circle $C_1$ with centre $P$. The line $4y + 3x = 31$ is

Question

$A(1, 7)$ and $B(9, 7)$ lie on a circle $C_1$ with centre $P$. The line $4y + 3x = 31$ is a normal to the circle and passes through $A$. Use a geometrical property of circle to explain why the $x$-coordinate of $P$ is 5.[2] Find the equation of the circle $C_1$.[3] The tangent to the circle at $A$ intersects the $y$-axis at $D$. Find the coordinates of $D$.[3] A larger circle $C_2$ has the same centre $P$ and radius three times that of $C_1$. $T$ is a point on $C_2$ such that $P$, $A$ and $T$ are collinear and the $x$-coordinate of $T$ is negative. Determine the coordinates of $T$.[2]

Accepted Answer

(i) $x_P=5$ (ii) $(x-5)^2+(y-4)^2=25$ (iii) $D=\left(0,\dfrac{17}{3}\right)$ (iv) $T=(-7,13)$

2025 CHS PRELIMS P2 Q10

2025 CHS PRELIMS P2 Q10

Solution:

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