A line has a slope -1 and passing through (2, 4). Find the point of in

1.	A line has a slope -1 and passing through (2, 4). Find the point of intersection of the line and its perpendicular line passing through (-1, 3)1. (1, 5)2. (1, 3)3. (1, 4)4. (1, 2)
Answer» Correct Answer - Option 1 : (1, 5) Concept: The general equation of a line is y = mx + c where m is the slope and c is any constant Slope of parallel lines are equal. Slope of perpendicular line have their product = -1 Equation of a line with slope m and passing through (x1, y1) (y - y1) = m (x - x1) Equation of a line passing through (x1, y1) and (x2, y2) is: \(\rm {y-y_1\over x-x_1}={y_2-y_1\over x_2-x_1}\) Calculation: The line has the slope -1 and passes through (2, 4) ∴ Equation of the perpendicular line is (y - y1) = m (x - x1) ⇒ y - 4 = -1 (x - 2) ⇒ y = -x + 6 ...(i) Slope(m1) = -1 and c1 = 6 Now for the slope of perpendicular line (m2) m1 × m2 = -1 ⇒ -1 × m2 = -1 ⇒ m2 = 1 Perpendicular line has the slope 1 and passes through (-1, 3) ∴ Equation of the perpendicular line is (y - y1) = m (x - x1) ⇒ y - 3 = 1 (x - (-1)) ⇒ y = x + 4 ...(ii) Adding (i) and (ii) ⇒ 2y = 10 ⇒ y = 5 Putting y in equation (ii) 5 = x + 4 ⇒ x = 1 ∴ The point of intersection is (1, 5)

A line has a slope -1 and passing through (2, 4). Find the point of intersection of the line and its perpendicular line passing through (-1, 3)1. (1, 5)2. (1, 3)3. (1, 4)4. (1, 2)

Answer» Correct Answer - Option 1 : (1, 5)

Concept:

The general equation of a line is y = mx + c

where m is the slope and c is any constant

Equation of a line with slope m and passing through (x1, y1)

(y - y1) = m (x - x1)

Equation of a line passing through (x1, y1) and (x2, y2) is:

\(\rm {y-y_1\over x-x_1}={y_2-y_1\over x_2-x_1}\)

Calculation:

The line has the slope -1 and passes through (2, 4)

∴ Equation of the perpendicular line is

(y - y1) = m (x - x1)

⇒ y - 4 = -1 (x - 2)

⇒ y = -x + 6 ...(i)

Slope(m1) = -1 and c1 = 6

Now for the slope of perpendicular line (m2)

m1 × m2 = -1

⇒ -1 × m2 = -1

⇒ m2 = 1

Perpendicular line has the slope 1 and passes through (-1, 3)

∴ Equation of the perpendicular line is

(y - y1) = m (x - x1)

⇒ y - 3 = 1 (x - (-1))

⇒ y = x + 4 ...(ii)

Adding (i) and (ii)

⇒ 2y = 10

⇒ y = 5

Putting y in equation (ii)

5 = x + 4

⇒ x = 1

∴ The point of intersection is (1, 5)