JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

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

Choose the correct answer:

Explanation

1. Let the Foot of the Perpendicular be $P(h, k)$

The point $P(h, k)$ lies on the line:

y = 3x + 4

So, it must satisfy the equation:

k = 3h + 4 \quad \dots \text{(1)}

2. Slope Relationship

The given line $L_1$ is $y = 3x + 4$ , so its slope $m_1 = 3$ .

The line $L_2$ passing through $(2, 3)$ and $(h, k)$ is perpendicular to $L_1$ .

The slope of $L_2$ is:

m_2 = \frac{k - 3}{h - 2}

Since $L_1 \perp L_2$ :

m_1 \cdot m_2 = -1

3 \cdot \left( \frac{k - 3}{h - 2} \right) = -1

3k - 9 = -h + 2

h + 3k = 11 \quad \dots \text{(2)}

3. Solving the Equations

Substitute (1) into (2):

h + 3(3h + 4) = 11

h + 9h + 12 = 11

10h = -1

h = -\frac{1}{10}

Now, find $k$ :

k = 3\left(-\frac{1}{10}\right) + 4

k = -\frac{3}{10} + \frac{40}{10}

k = \frac{37}{10}

Final Result

The coordinates of the foot of the perpendicular are:

\left( -\frac{1}{10}, \frac{37}{10} \right)