Q34. - Geometry

Question

 Find two values of k such that the points (-3, 4), (0, k), and (k, 10) are collinear.

Step-by-Step Solution

Answer

The two values of k are$-5,6$ .

Step-1 – Given

The given points are$(-3,4),(0,k)\text{and}(k,10)$ .

Step-2 – To determine

We have to find two values of k such that the points$(-3,4),(0,k)\text{and}(k,10)$ are collinear.

Step-3 – Calculation

The slope formula for two points$(x_1,y_1)\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}(x_2,y_2)$ is:$m=\frac{y_2-y_1}{x_2-x_1}$ .

The given points are$A(-3,4),B(0,k)\text{and }C(k,10)$ .

Since the given points are collinear so:

$\begin{array}{l}{m}_{AB}=m_{BC}=m_{AC}\\ \frac{k-4}{0-(-3)}=\frac{10-k}{k-0}=\frac{10-4}{k-(-3)}\end{array}$

Solving this equation, we get:

 $\begin{array}{l}{m}_{AB}=m_{BC}\\ \frac{k-4}{0-(-3)}=\frac{10-k}{k-0}\\ \frac{k-4}{3}=\frac{10-k}{k}\\ k(k-4)=3(10-k)\\ k^2-4k=30-3k\\ k^2-k-30=0\\ k^2-6k+5k-30=0\\ k(k-6)+5(k-6)=0\\ (k-6)(k+5)=0\\ k-6=0\\ k=6\\ k+5=0\\ k=-5\end{array}$

So, the two values of k are $-5,6$.