Suppose that the points (h,k), (1,2) and (-3,4) lie on the line L_1.

Question

Suppose that the points $(h,k), (1,2)$ and $(-3,4)$ lie on the line $L_1$. If a line $L_2$ passing through the points $(h,k)$ and $(4,3)$ is perpendicular on $L_1$, then $\dfrac kh$ equals:

Accepted Answer

Equation of line L1 passing through points (1, 2) and (--3, 4) is :

y - 2 = $$\left( {{{4 - 2} \over { - 3 - 1}}} \right)(x - 1)

\[ \Rightarrow $$ y - 2 = $$ - {1 \over 2}(x - 1)

\[ \Rightarrow $$ 2y -- 4 = --x + 1

$$ \Rightarrow $$ x + 2y = 5 .......(L1)

Line L1 passes through point (h, k).

$$ \therefore $$ h + 2k = 5 .......(1)

Line L2 is perpendicular to line L1.

$$ \therefore $$ Equation of line L2 :

2x - y = $$\lambda $$

This line L2 passes through point (4, 3).

$$ \therefore $$ 2(4) - 3 = $$\lambda $$

$$ \Rightarrow $$ $$\lambda $$ = 5

$$ \therefore $$ Line L2 :

2x - y = 5

L2 also passes through (h, k)

$$ \therefore $$ 2h - k = 5  ............(2)

Solving (1) and (2) we get,

h = 3 and k = 1

$$ \therefore $$ $${k \over h} = {1 \over 3}$$

Solutions · 1