Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | January | Q#3

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Question

The line L has equation y=5 – 2x.

a.   Show that the point P (3, –1) lies on L.

b.   Find an equation of the line perpendicular to L, which passes through
P. Give your answer in the form ax + by + c = 0, where a, b and c are integers.

Solution

a.
 

If a point lies on the curve (or the line), the coordinates of that point satisfy the equation of the curve  (or the line).

We are given equation of the line L as;

We are also given coordinates of the point P(3,-1).

We substitute x-coordinate of point P in the equation of the curve L;

Since coordinates of the point P satisfy the equation of the line L, therefore the point P(3,-1) lies on  the line.

b.
 

We are required to find the equation of the line  which passes through point P(3,-1) and is  perpendicular to line L with equation .

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have coordinates of a point on the line  as P(3,-1).

We need to find slope of the line .

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is; 

Therefore, if we can find slope of the line L then we can find slope of the line .

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

We are given equation of line L and we can write it in slope-intercept form as follows;

Hence, slope of line L is .

Hence, slope of line  can be found as;

Point-Slope form of the equation of the line is;

Now with coordinates of a point P(3,-1) on the line  and slope , we can write equation of the line .

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