Question

If the slope of a line passing through the point $A(3,2)$ is $\frac{3}{4}$, then which of the following are the coordinates of the point lie on the line which are 5 units away from the point $A$ ?

• A. $(-1,-1)$
• B. $(7,5)$
• C. Both(a) and (b)
• D. None of these

Answer 1

Answer: (C)

Equation of a line passing through $(3,2)$ having slope $\frac{3}{4}$ is given by
$ y-2 =\frac{3}{4}(x-3) $
$ \Rightarrow 4 y-8 =3 x-9 $
$\Rightarrow 4 y-3 x+1 =0.......$(i)
Let $(h, k)$ be the point on the line which is 5 units away from the point $A$. Then,
$\sqrt{(h-3)^2+(k-2)^2}=5 ......$(ii)
Also, we have
$4 k-3 h+1 =0$ [from Eq. (i)]
$\Rightarrow k =\frac{3 h-1}{4} ......$(iii)
Putting the value of $k$ in Eq. (ii) we get
$ \sqrt{(h-3)^2+\left(\frac{3 h-1}{4}-2\right)^2} =5$
$ \Rightarrow 16(h-3)^2+(3 h-9)^2 =400 $
$ \Rightarrow 16 h^2+144-96 h+9 h^2+81-54 h =400 $
$ \Rightarrow 25 h^2-150 h-175 =0$
$\Rightarrow h^2-6 h-7=0 $
$\Rightarrow (h+1)(h-7)=0$
$\Rightarrow h=-1, h=7$
Putting these value of $h$ in Eq. (iii), we get
$ k=\frac{3(-1)-1}{4}$
$\Rightarrow k=-1 $
and $ k=\frac{3(7)-1}{4}$
$\Rightarrow k=5$
Therefore, the coordinates of the required points are either $(-1,-1)$ or $(7,5)$.