Find the values of k for which points A(-6, 10), (B(24, k) and C(3, 28) are collinear Also, - Maths - Co-ordinate Geometry

Question

Find the values of k for which points A(-6, 10), (B(24, k) and
C(3, 28) are collinear Also, find the ratio in which B divides AC
and the length of AC

Manbar Singh · Accepted Answer

If the points A, B and C are collinear then the area of the ∆
formed by these points = 0

or 12 x1 y2 - y3 + x2 y3 - y1 + x3 y1 - y2 = 0⇒12 -6 k - 28 +
24 28 - 10 + 3 10 - k = 0⇒12 -6k + 168 +24 × 18 + 30 -
3k = 0⇒12 -6k + 168 + 432 + 30 -3k = 0⇒12 -9k + 630 =
0⇒-9k + 630 = 0⇒-9k = -630⇒ k = 70Let the point B
24, 70 divides the line segment joining the point A -6, 10 and the
point C 3, 28 in the ratio m1:m2then x= 24, y = 70x1 = -6, y1 =
10x2 = 3, y2 = 28Now applying the section formulax =m1 x2 + m2 x1m1
+ m2⇒24 = m1 × 3 + m2 ×-6m1 + m2⇒24 m1 + 24
m2 = 3 m1 - 6 m2⇒24 m1 - 3 m1 = -6 m2 - 24 m2⇒21 m1 =
-30 m2⇒m1m2 = -3021⇒m1m2 = -107m1 : m2 = 10 : 7
externallyTherefore point B divides AC in the ratio 10 : 7
externallyThe length of AC =x2 - x12 + y2 - y12= 3 + 62 + 28 - 102=
92 + 182= 81 + 144= 225 = 15 units

Find the values of k for which points A(-6, 10), (B(24, k) and C(3, 28) are collinear . Also, find the ratio in which B divides AC and the length of AC.