Mathematics

The line segment joining A(2, -3) and B(-3, 2) is intercepted by the x-axis at the point M and the y-axis at the point N. PQ is perpendicular to AB at R and meets the y-axis at a distance of 6 units form the origin O, as shown in the diagram, at S. Find the :
(a) coordinates of M and N.
(b) coordinates of S
(c) slope of AB.
(d) equation of line PQ.

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(a) From figure,

Coordinates of M = (-1, 0) and coordinates of N = (0, -1).

(b) From figure,

Coordinates of S = (0, 6).

Slope = $\dfrac{y$

Slope of AB = $\dfrac{2 - (-3)}{-3 - 2} = \dfrac{2 + 3}{-5} = \dfrac{5}{-5}$ = -1.

Hence, slope of AB = -1.

(d) We know that,

Product of slope of perpendicular lines = -1.

∴ Slope of AB × Slope of PQ = -1

⇒ -1 × Slope of PQ = -1

⇒ Slope of PQ = $\dfrac{-1}{-1}$ = 1.

Since, PQ passes through point S.

By point-slope form,

Equation of line : y - y₁ = m(x - x₁)

⇒ y - 6 = 1(x - 0)

⇒ y - 6 = x

⇒ y = x + 6.

Hence, equation of line PQ is y = x + 6.

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