Question

The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.

पर्याय

• ${\displaystyle \frac{37}{10},\frac{-1}{10}}$

• ${\displaystyle \frac{-1}{10},\frac{37}{10}}$

• ${\displaystyle \frac{10}{37},-10}$

• ${\displaystyle \frac{2}{3},-\frac{1}{3}}$

Answer 1

The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ${\displaystyle \frac{-1}{10},\frac{37}{10}}$.

Explanation:

Given equation is y = 3x + 4   .....(i)

⇒ 3x – y + 4 = 0

Slope = 3

Equation of any line passing through the point (2, 3) is

 y – 3 = m(x – 2)   .....(ii)

If equation (i) is perpendicular to eq. (ii)

Then m × 3 = – 1    ......${\displaystyle [\because m_1\times m_2=-1]}$

⇒ m = ${\displaystyle -\frac{1}{3}}$

Putting the value of m in equation (ii) we get

y – 3 = ${\displaystyle -\frac{1}{3}(x-2)}$

⇒ 3y – 9 = – x + 2

⇒ x + 3y = 11  .....(iii)

Solving equation (i) and equation (iii) we get

3x – y = – 4

⇒ y = 3x + 4   ......(iv)

Putting the value of y in eq. (iii) we get

x + 3(3x + 4) = 11

⇒ x + 9x + 12 = 11

⇒ 10x = – 1

⇒ x = ${\displaystyle \frac{-1}{10}}$

From equation (iv) we get

y = ${\displaystyle 3\left(\frac{-1}{10}\right)+4}$

⇒ y = ${\displaystyle \frac{-3}{10}+4}$

⇒ y = ${\displaystyle \frac{37}{10}}$

So the required coordinates are ${\displaystyle (\frac{-1}{10},\frac{37}{10})}$.