(a) Solve for x and y in the following equations : \(2x - y = \frac{9}{2}\) \(x + 4y = 0\) (b) In the diagram, TA is a tangent to the circle at A. If \(\sta...
Question 1 Report
(a) Solve for x and y in the following equations :
\(2x - y = \frac{9}{2}\)
\(x + 4y = 0\)
(b) 
In the diagram, TA is a tangent to the circle at A. If \(\stackrel\frown{BCA} = 40°\) and \(\stackrel\frown{DAT} = 52°\), find \(\stackrel\frown{BAD}\).
Answer Details
(a) Simultaneous equations.
\[2x-y=\frac{9}{2}\quad(1),\qquad x+4y=0\quad(2).\]
From (2), \(x=-4y\). Substitute into (1):
\[2(-4y)-y=\frac{9}{2}\]
\[-8y-y=\frac{9}{2}\]
\[-9y=\frac{9}{2}\ \Rightarrow\ y=-\frac{1}{2}.\]
Then \(x=-4y=-4\left(-\dfrac{1}{2}\right)=2\).
\[\boxed{x=2,\quad y=-\tfrac{1}{2}}.\]
(b) Tangent TA to the circle at A.
From the diagram, \(B, C, D, A\) lie on the circle, \(TA\) is the tangent at \(A\), \(\angle BCA=40^\circ\) (at \(C\)) and \(\angle DAT=52^\circ\) (between the tangent \(AT\) and chord \(AD\)).
Chord AB. By the alternate segment theorem, the tangent-chord angle between the tangent (on the side towards \(X\), opposite \(T\)) and chord \(AB\) equals the angle in the alternate segment \(\angle BCA\):
\[\angle BAX=\angle BCA=40^\circ.\]
Straight tangent line. The points \(X, A, T\) lie on the straight tangent line, so the three angles at \(A\) on one side sum to \(180^\circ\):
\[\angle BAX+\angle BAD+\angle DAT=180^\circ.\]
\[40^\circ+\angle BAD+52^\circ=180^\circ\]
\[\angle BAD=180^\circ-40^\circ-52^\circ=\boxed{88^\circ}.\]
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