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Mathematics

Assertion (A) : The x-coordinate of the point which divides the line segment joining points (6, 3) and (-4, 5) in the ratio 3 : 2 internally is 0.

Reason (R) : The coordinates of point P dividing the line segment joining the points A(x1, y1) and B(x2, y2) in the ratio m : n is

(mnx2+x1mn+1,mny2+y1mn+1)\Big(\dfrac{\dfrac{m}{n}x2 + x1}{\dfrac{m}{n} + 1}, \dfrac{\dfrac{m}{n}y2 + y1}{\dfrac{m}{n} + 1}\Big)

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

Section Formula

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Answer

By section-formula,

(x, y) = m1x2+m2x1m1+m2\dfrac{m1x2 + m2x1}{m1 + m2}

Substituting values we get :

(x,y)=(3×4+2×63+2,3×5+2×33+2)=(12+125,15+65)=(05,215)=(0,215).\Rightarrow (x, y) = \Big(\dfrac{3 \times -4 + 2 \times 6}{3 + 2}, \dfrac{3 \times 5 + 2 \times 3}{3 + 2}\Big) \\[1em] = \Big(\dfrac{-12 + 12}{5}, \dfrac{15 + 6}{5}\Big) \\[1em] = \Big(\dfrac{0}{5}, \dfrac{21}{5}\Big) \\[1em] = \Big(0, \dfrac{21}{5}\Big).

∴ Assertion (A) is true.

Solving,

(mx2+nx1nm+nn,my2+ny1nm+nn)(mx2+nx1m+n,my2+ny1m+n).\Rightarrow \Big(\dfrac{\dfrac{mx2 + nx1}{n}}{\dfrac{m + n}{n}}, \dfrac{\dfrac{my2 + ny1}{n}}{\dfrac{m + n}{n}}\Big) \\[1em] \Rightarrow \Big(\dfrac{mx2 + nx1}{m + n}, \dfrac{my2 + ny1}{m + n}\Big).

∴ Reason (R) is true.

Hence, Option 3 is the correct option.

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