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5.2 · Q28

Q.If the points A(3,0,p)(3,0,p), B(−1,q,3)(-1,q,3) and C(−3,3,0)(-3,3,0) are collinear, then find the ratio in which the point C divides the line segment AB.

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Let A(3,0,p),B(−1,q,3),C(−3,3,0)A(3,0,p),B(-1,q,3),C(-3,3,0), and suppose CC divides ABAB in the ratio t:1t:1, so

C=tB+At+1.C=\frac{tB+A}{t+1}.

Using the x-coordinate (which involves no unknowns besides tt): −3=t(−1)+3t+1-3=\dfrac{t(-1)+3}{t+1}, so

−3(t+1)=−t+3⇒−3t−3=−t+3⇒−2t=6⇒t=−3-3(t+1)=-t+3\Rightarrow -3t-3=-t+3\Rightarrow -2t=6\Rightarrow t=-3. …

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