Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, P be a point on side BC, and $\Delta_{1}$ and $\Delta_{2}$ be the areas of triangle APB and ABC respectively. If $\Delta_{1}: \Delta_{2}=4:7$, then the area enclosed by the lines AP, AC and the x-axis is

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Straight Lines

Let

A (1, 1)

B (- 4, 3)

C (- 2, - 5)

be vertices of a triangle

A B C

, P be a point on side BC, and

Δ_{1}

and

Δ_{2}

be the areas of triangle APB and ABC respectively.

If

Δ_{1} : Δ_{2} = 4 : 7

, then the area enclosed by the lines AP, AC and the x-axis is

Answer: (C)

Using the area ratio

\frac{Δ _{1}}{Δ _{2}} = \frac{4}{7}

and determinant formulas for areas, we find that point P divides BC in ratio 4:3.

The area of triangle ABC is

Δ_{2} = \frac{1}{2} ∣1 (3 + 5) + (- 4) (- 5 - 1) + (- 2) (1 - 3) ∣ = \frac{1}{2} ∣8 + 24 + 4∣ = 18

.

Using the ratio condition and solving for P's coordinates on line BC (equation:

4 x + y = - 13

), we get

P (\frac{- 20}{7}, \frac{- 11}{7})

.

The line AP intersects the

x

-axis at

Q (\frac{- 1}{2}, 0)

and AC intersects at

R (\frac{1}{2}, 0)

.

Area of triangle AQR =

\frac{1}{2} \times 1 \times 1 = \frac{1}{2}

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option