NIMCET 2015 — Mathematics PYQ

1. Using the Foot of Perpendicular Formula

For a point $(x_1,y_1)$ and a line $ax+by+c=0$, the coordinates of the foot of the perpendicular $(h,k)$ are given by:

2. Identify the Variables

• Point $(x_1,y_1)=(2,4)$

• Line equation: $x+y-1=0⟹a=1,b=1,c=-1$

3. Substitute into the Formula

First, calculate the constant ratio:

Now, set up the equations for $h$ and $k$:

• For $h$:

  $$\frac{h-2}{1}=-\frac{5}{2}$$
  $$h=2-\frac{5}{2}=\frac{4-5}{2}=-\frac{1}{2}$$

• For $k$:

  $$\frac{k-4}{1}=-\frac{5}{2}$$
  $$k=4-\frac{5}{2}=\frac{8-5}{2}=\frac{3}{2}$$

4. Final Coordinates

The foot of the perpendicular is $(-\frac{1}{2},\frac{3}{2})$.

Tip: You can verify this by checking if the point lies on the line.

Substituting into $x+y=1$:

$-\frac{1}{2}+\frac{3}{2}=\frac{2}{2}=1$. The condition is satisfied.

Correct Option: (b)