JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Straight Lines

Consider a triangle having vertices

A (- 2, 3), B (1, 9)

and

C (3, 8)

. If a line L passing through the circum-center of triangle

A B C

, bisects line

B C

, and intersects

y

-axis at point

(0, \frac{α}{2})

, then the value of real number

α

is ______

Answer: 9

Solution

A line through the circumcentre that bisects

B C

is the perpendicular bisector of

B C

, since the circumcentre is equidistant from

B

and

C

. The midpoint of

B (1, 9)

and

C (3, 8)

(2, \frac{17}{2})

, and the slope of

B C

\frac{8 - 9}{3 - 1} = - \frac{1}{2}

, so the perpendicular bisector has slope

2

y - \frac{17}{2} = 2 (x - 2) ⟹ y = 2 x + \frac{9}{2}

x = 0

this gives

y = \frac{9}{2}

, so

\frac{α}{2} = \frac{9}{2}

and

α = 9

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Numerical answer

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