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Circle Geometry, SMB-016

The line \(BD\) is a tangent to a circle and the secant \(AD\) intersects the circle at \(A\) and \(C\).
 

Given that  \(AC = 18\)  and  \(CD = 6\), find the value of \(x\).  (2 marks)   

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\(x=12\)

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\(x^2\) \(= 6 \times (18 + 6) \)  
  \(=144\)  
\(x\) \(=12\)  

Circle Geometry, SMB-002

The line \(AT\) is the tangent to the circle at \(A\), and \(BT\) is a secant meeting the circle at \(B\) and \(C\).
  

Given that  \(AT = 12\),  \(BC = 7\)  and  \(CT = x\), find the value of \(x\).  (2 marks)

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\(x = 9\)

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\(\text{Property: square of tangent = product of secant intercepts}\)

\(AT^2\) \(= CT \times BT\)
\(12^2\) \(= x(x + 7)\)
\(144\) \(= x^2 + 7x\)
\(x^2 + 7x-144\) \(= 0\)
\((x + 16)(x-9)\) \(= 0\)

 

\(\therefore x = 9,\  (x \gt 0) \)

Plane Geometry, EXT1 2018 HSC 11d

Two secants from the point `C` intersect a circle as shown in the diagram.
 

 
What is the value of `x`?  (2 marks)

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`4`

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`text(Using formula for intercepts of intersecting secants:)`

`x (x + 2)` `= 3 (3 + 5)`
`x^2 + 2x` `= 24`
`x^2 + 2x – 24` `= 0`
`(x + 6) (x – 4)` `= 0`
`:. x` `= 4 \ \ \ (x > 0)`

Plane Geometry, EXT1 2015 HSC 3 MC

Two secants from the point `P` intersect a circle as shown in the diagram.
  

What is the value of `x`?

  1. `2`
  2. `5`
  3. `7`
  4. `8`
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`B`

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`text{Property: products of intercepts of secants from external point are equal}`

`x(x + 3)` `= 4(4 + 6)`
`x^2 + 3x` `= 40`
`x^2 + 3x-40` `= 0`
`(x-5)(x + 8)` `= 0`

 
`:.x = 5,\ \  (x>0)`

`=>B`