Find the equation of AB and calculate |AB|, leaving your answer in the form a√b. Points A are (−4, 3) and B…

Question

Find the equation of AB and calculate |AB|, leaving your answer in the form a√b. Points A are (−4, 3) and B are (2, −6).

ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 6a: a) i) Find the equation of AB. Step 1: Calculate the slope m of the line passing through A (−4, 3) and B (2, −6). m = (y₂ − y₁)/(x₂ − x₁) = (−6 − 3)/(2 − (−4)) = (−9)/(2 + 4) = (−9)/(6) = −3/2; Step 2: Use the point-slope form y − y₁ = m(x − x₁) with point A (−4, 3) and m = −3/2. y − 3 = −3/2(x − (−4)); y − 3 = −3/2(x + 4); Step 3: Convert the equation to the general form Ax + By + C = 0. 2(y − 3) = −3(x + 4); 2y − 6 = −3x − 12; 3x + 2y − 6 + 12 = 0; 3x + 2y + 6 = 0; The equation of AB is 3x + 2y + 6 = 0. ii) Calculate |AB|, leaving your answer in the form a√(b). Step 1: Use the distance formula D = √((x₂ − x₁)² + (y₂ − y₁)²) for points A (−4, 3) and B (2, −6). |AB| = √((2 − (−4))² + (−6 − 3)²); |AB| = √((6)² + (−9)²) Step 2: Simplify the expression. |AB| = √(36 + 81); |AB| = √(117) Step 3: Express √(117) in the form a√(b) by finding the largest perfect square factor of 117. 117 = 9 × 13; |AB| = √(9 × 13) = √(9) × √(13) = 3√(13) The length |AB| is 3√(13).

Find the equation of AB and calculate |AB|, leaving your answer in the form a√b. Points A are (-4, 3) and B are (2, -6).

Step-by-step solution

Part a(i)· 3 steps

Calculate the slope $m$ of the line passing through A $(-4, 3)$ and B $(2, -6)$ .

Use the point-slope form $y - y_1 = m(x - x_1)$ with point A $(-4, 3)$ and $m = -\frac{3}{2}$ .

Convert the equation to the general form $Ax + By + C = 0$ .

Part a(ii)· 3 steps

Use the distance formula $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for points A $(-4, 3)$ and B $(2, -6)$ .

Simplify the expression.

Express $\sqrt{117}$ in the form $a\sqrt{b}$ by finding the largest perfect square factor of 117.

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