Class 10

Math

All topics

Coordinate Geometry

In each of the following find the value of $k$, for which the points are collinear.

(i) $(7,−2),(5,1),(3,k)$

(ii) $(8,1),(k,−4),(2,−5)$

These point are collinear if area ($△ABC$)=0

$⇒x_{1}(y_{2}−y_{3})+x_{2}(y_{3}−y_{1})+x_{3}(y_{1}−y_{2})=0$

Here $x_{1},y_{1})=(7.−2)$

$(x_{2},y_{2})=(5,1)$

$(x_{3},y_{3})=(3,k)$

$⇒7(1−k)+5(k+2)+3(−2−1)=0$

$⇒7−7k+5k+10−9=0$

$⇒8−2k=0$

$⇒2k=8$

$⇒k=4$

Hence the given points are collinear for $k=4$

(i) let the given points are $A(8,1$),$B(k,−4)$ and $C(2,−5)$

These point are collinear if area ($△ABC$)=0

$⇒x_{1}(y_{2}−y_{3})+x_{2}(y_{3}−y_{1})+x_{3}(y_{1}−y_{2})=0$

Here $x_{1},y_{1})=(8,1)$

$(x_{2},y_{2})=(k,−4)$

$(x_{3},y_{3})=(2,−5)$

$⇒8(−4+5)+k(−5−1)+2(1+4)=0$

$⇒8−6k+10=0$

$⇒−6k=−18$

$⇒k=3$

$⇒k=4$

Hence the given points are collinear for $k=3$