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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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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\(x=12\)
\(x^2\) | \(= 6 \times (18 + 6) \) | |
\(=144\) | ||
\(x\) | \(=12\) |
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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\(x = 9\)
\(\text{Property: square of tangent = product of secant intercepts}\)
\(AT^2\) | \(= CT \times BT\) |
\(12^2\) | \(= x(x + 7)\) |
\(144\) | \(= x^2 + 7x\) |
\(x^2 + 7x-144\) | \(= 0\) |
\((x + 16)(x-9)\) | \(= 0\) |
\(\therefore x = 9,\ (x \gt 0) \)
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`
`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)` |
Two secants from the point `P` intersect a circle as shown in the diagram.
What is the value of `x`?
`B`
`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`