A line segment joins the points A(1, 4) and B(7, 10). Point P divides the line segment AB … — JAMB UTME

A line segment joins the points $A(1, 4)$ and $B(7, 10)$ . Point $P$ divides the line segment $AB$ internally in the ratio $1:2$ . A second line $L_2$ passes through $P$ and is perpendicular to $AB$ . If line $L_2$ intersects the y-axis at the point $(0, k)$ , find the value of $k$ .

Cevap: 9

Cevap

The value of k is 9.

Using the section formula for internal division in a 1:2 ratio, the coordinates of point P are found to be (3, 6). The gradient of the segment AB is 1, which means the perpendicular line L_2 has a gradient of -1. Writing the equation of line L_2 passing through (3, 6) yields y = -x + 9. Evaluating at x = 0 gives the y-intercept k = 9.

Adım Adım Çözüm

Calculate the coordinates of point P dividing segment AB internally in the ratio 1:2.

P = (3, 6)

Using the section formula x = (m x_2 + n x_1) / (m + n) and y = (m y_2 + n y_1) / (m + n) with ratio m:n = 1:2.

Calculate the gradient m_1 of the line segment AB.

m_1 = 1

Applying the gradient formula m = (y_2 - y_1) / (x_2 - x_1) gives (10 - 4) / (7 - 1) = 1.

Determine the gradient m_2 of the perpendicular line L_2.

m_2 = -1

Perpendicular lines satisfy m_1 * m_2 = -1, hence m_2 = -1 / 1 = -1.

Find the equation of line L_2 passing through P(3, 6) with gradient -1.

y = -x + 9

Using point-slope form y - y_1 = m(x - x_1) yields y - 6 = -1(x - 3).

Determine the y-intercept value k by setting x = 0.

k = 9

Substituting x = 0 into y = -x + 9 gives y = 9.

Anahtar Kavram

Section formula, perpendicular line gradients, and y-intercept determination

Tahmini Süre:3m 0s

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