(a) In the diagram, A, B, C and D are points on the circumference of a circle. XY is a tangent at A. Find : (i) < CAX ; (ii) < ABY. (b) If (m + 1) and (m - ...

Answer Details

(a) Circle through A, B, C, D with tangent XY at A.

From the diagram: \(\angle ADB=20^\circ\) (at D), and the tangent-secant angle at Y (between tangent \(YA\) and the secant through \(B\) and \(C\)) is \(\angle AYC=69^\circ\), with \(C,\,B,\,Y\) in a straight line.

(i) \(\angle CAX\).

\(\angle ADB=20^\circ\) is an inscribed angle on chord \(AB\), so

\[\text{arc } AB=2\times20^\circ=40^\circ.\]

The tangent-chord angle \(\angle BAY\) equals the angle in the alternate segment \(\angle ADB\):

\[\angle BAY=20^\circ.\]

In triangle \(ABY\), \(\angle AYB=69^\circ\), so

\[\angle ABY=180^\circ-69^\circ-20^\circ=91^\circ,\]

and since \(C,B,Y\) are collinear,

\[\angle ABC=180^\circ-\angle ABY=180^\circ-91^\circ=89^\circ.\]

By the alternate segment theorem, the tangent-chord angle \(\angle CAX\) equals the inscribed angle \(\angle ABC\) in the alternate segment:

\[\angle CAX=\angle ABC=\boxed{89^\circ}.\]

(ii) \(\angle ABY\).

From the working above,

\[\angle ABY=180^\circ-69^\circ-20^\circ=\boxed{91^\circ}.\]

(b) Factors of \(m^{2}-km+c\).

If \((m+1)\) and \((m-3)\) are factors, then

\[m^{2}-km+c=(m+1)(m-3)=m^{2}-3m+m-3=m^{2}-2m-3.\]

Comparing coefficients:

\[-k=-2\ \Rightarrow\ \boxed{k=2},\qquad c=\boxed{-3}.\]