# If the Points A(−2, 1), B(A, B) and C(4, −1) Ae Collinear and a − B = 1, Find the Values of a and B. - Mathematics

If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.

#### Solution

The given points A(−2, 1), B(ab) and C(4, −1) are collinear.

$\therefore \text{ ar } \left( ∆ ABC \right) = 0$
$\Rightarrow \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right| = 0$
$\Rightarrow x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) = 0$

$\Rightarrow - 2\left[ b - \left( - 1 \right) \right] + a\left( - 1 - 1 \right) + 4\left( 1 - b \right) = 0$

$\Rightarrow - 2b - 2 - 2a + 4 - 4b = 0$

$\Rightarrow - 2a - 6b = - 2$

$\Rightarrow a + 3b = 1 . . . . . \left( 1 \right)$

Also, it is given that

a − b = 1               .....(2)

Solving (1) and (2), we get

$b + 1 + 3b = 1$
$\Rightarrow 4b = 0$
$\Rightarrow b = 0$

Putting b = 0 in (1), we get

$a + 3 \times 0 = 1$

$\Rightarrow a = 1$

Hence, the respective values of a and b are 1 and 0.

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.5 | Q 32 | Page 55