JEE MAIN - Mathematics (2022 - 27th July Morning Shift - No. 13)
Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ and $$\Delta_{2}$$ be the areas of triangles $$A P B$$ and $$A B C$$, respectively. If $$\Delta_{1}: \Delta_{2}=4: 7$$, then the area enclosed by the lines $$A P, A C$$ and the $$x$$-axis is :
$$\frac{1}{4}$$
$$\frac{3}{4}$$
$$\frac{1}{2}$$
1
Explanation
$${{{\Delta _1}} \over {{\Delta _2}}} = {{{1 \over 2} \times BP \times AH} \over {{1 \over 2} \times BC \times AH}} = {4 \over 7}$$
_27th_July_Morning_Shift_en_13_1.png)
$$P\left( {{{ - 20} \over 7},{{ - 11} \over 7}} \right)$$
Line $$AC:y - 1 = 2(x - 1)$$
Intersection with x-axis $$ = \left( {{1 \over 2},0} \right)$$
Line $$AP:y - 1 = {2 \over 3}(x - 1)$$
Intersection with x-axis $$\left( {{{ - 1} \over 2},0} \right)$$
Vertices are $$(1,1),\left( {{1 \over 2},0} \right)$$ and $$\left( {{{ - 1} \over 2},0} \right)$$
Area $$ = {1 \over 2}$$ sq. unit
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