Question

Two points (5, 2) and (2, a). Line passes through these points makes an angle of π/4 at origin. The product of all values of a is equal to

• A. 8
• B. -4
• C. -2
• D. 1

Answer 1

The Correct Option is B

To solve this problem, we need to find the value of \( a \) that makes the line passing through points (5, 2) and (2, a) create an angle of \( \frac{\pi}{4} \) with the positive x-axis at the origin. The line has a slope, which dictates this angle, and we need to use this relationship to find the possible values of \( a \) and subsequently the product of these values.

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

For our line, plugging in the points (5, 2) and (2, a), the slope becomes:

\(m = \frac{a - 2}{2 - 5} = \frac{a - 2}{-3}\) 

The line makes an angle of \( \frac{\pi}{4} \) at the origin. The tangent of this angle (i.e., the slope of the line) is \(\tan(\frac{\pi}{4}) = 1\).

Thus, we set the slope to be equal to 1:

\(\frac{a - 2}{-3} = 1\)

Solving for \( a \):

\(a - 2 = -3\)

\(a = -1\)

Thus, \((-1)\) is one solution for \( a \).

However, the line can also make an angle of \(\frac{\pi}{4}\) in the opposite direction, i.e., with a negative slope of \(-1\), for which:

\(\frac{a - 2}{-3} = -1\)

Solving for \( a \) in this case:

\(a - 2 = 3\)

\(a = 5\)

Thus, \( 5 \) is the second solution for \( a \).

Therefore, the two possible values of \( a \) are \(-1\) and \(5\).

The product of all values of \( a \) is:

\(-1 \times 5 = -5\)

As per the given correct answer

-4

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