NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

A straight line through the point (4, 5) is such that its intercept between the axes is bisected at A, then its equation is

Choose the correct answer:

Explanation

To find the equation of the line, we can use the Intercept Form of a straight line.

1. Define the Intercepts

Let the line intersect the $x$ -axis at $P(a, 0)$ and the $y$ -axis at $Q(0, b)$ .

The equation of the line in intercept form is:

\frac{x}{a} + \frac{y}{b} = 1

2. Use the Midpoint Formula

The problem states that the point $A(4, 5)$ bisects the segment $PQ$ . According to the midpoint formula, the coordinates of the midpoint are:

\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

For points $P(a, 0)$ and $Q(0, b)$ , the midpoint is:

\left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = (4, 5)

By comparing the coordinates:

$\frac{a}{2} = 4 \implies a = 8$
$\frac{b}{2} = 5 \implies b = 10$

3. Formulate the Equation

Now, substitute the values of $a = 8$ and $b = 10$ back into the intercept form:

\frac{x}{8} + \frac{y}{10} = 1

To convert this into the general form, find the Least Common Multiple (LCM) of 8 and 10, which is 40. Multiply the entire equation by 40:

40 \left( \frac{x}{8} \right) + 40 \left( \frac{y}{10} \right) = 40(1)

5x + 4y = 40

Final Answer:

The equation of the line is $5x + 4y = 40$ . The correct option is D.