Question

for which value of m the lines y=mx+2 and y=7x+5 are perpendicular
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Answer 1

$\large\underline{\sf{Solution-}}$

Given equation of lines are

$\rm :\longmapsto\:y = mx + 2 - - - - (1)$

and

$\rm :\longmapsto\:y = 7x + 5 - - - - (2)$

We know,

The equation of line of the form, y = mx + c provides the information that m is the slope of line and c is the length of intercept made by the line on y - axis.

So, y = mx + 2 represents the equation of line having slope m.

and

y = 7x + 5 represents the equation of line having slope 7.

Further, it is given that line (1) and (2) are perpendicular to each other.

We know,

Two lines having slope m and M are perpendicular iff Mm = - 1.

$\rm \implies\:7m \: = \: - \: 1$

$\rm \implies\:\boxed{ \tt{ \: m \: = \: - \: \frac{1}{7} \: }}$

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More to know :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.

Answer 2

Answer:

$\large\underline{\sf{Solution-}}$

Given equation of lines are

$\rm :\longmapsto\:y = mx + 2 - - - - (1)$

and

$\rm :\longmapsto\:y = 7x + 5 - - - - (2)$

We know,

The equation of line of the form, y = mx + c provides the information that m is the slope of line and c is the length of intercept made by the line on y - axis.

So, y = mx + 2 represents the equation of line having slope m.

and

y = 7x + 5 represents the equation of line having slope 7.

Further, it is given that line (1) and (2) are perpendicular to each other.

We know,

Two lines having slope m and M are perpendicular iff Mm = - 1.

$\rm \implies\:7m \: = \: - \: 1$

$\rm \implies\:\boxed{ \tt{ \: m \: = \: - \: \frac{1}{7} \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More to know :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.