Here is a sketch of the line L. The points P(-6, 0) and Q(0, 3) are points on the line L. The point R is such that PQR is a straight line and PQ : QR = 2 : 3. (a) Find the coordinates of R. (b) Find an equation of the line that is perpendicular to L and passes through Q.

GCSE Edexcel Maths Expanding Brackets: Here is a sketch of the line L.

Here is a sketch of the line L - Edexcel - GCSE Maths - Question 12 - 2021 - Paper 2

Question 12

Here is a sketch of the line L. The points P(-6, 0) and Q(0, 3) are points on the line L. The point R is such that PQR is a straight line and PQ : QR = 2 : 3. (a) ... show full transcript

Worked Solution & Example Answer:Here is a sketch of the line L - Edexcel - GCSE Maths - Question 12 - 2021 - Paper 2

Step 1

Find the coordinates of R.

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Answer

To find the coordinates of R, we first calculate the distance segment PQ.

The coordinates of P are (-6, 0) and Q are (0, 3). Using the distance formula, we calculate:

$d_{PQ} = \sqrt{(0 - (-6))^2 + (3 - 0)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}$

Given the ratio PQ : QR = 2 : 3, we can let the length of PQ be 2x and QR be 3x.

Thus, we have:

$2x + 3x = d_{PQ} \Rightarrow 5x = 3\sqrt{5} \Rightarrow x = \frac{3\sqrt{5}}{5}$

This gives:

Length of PQ = $2x = \frac{6\sqrt{5}}{5}$
Length of QR = $3x = \frac{9\sqrt{5}}{5}$

Now, we can find point R using the section formula, where R divides PQ in the ratio 2:3:

R's coordinates can be found by:

$R = \left( \frac{3 \cdot 0 + 2 \cdot (-6)}{2 + 3}, \frac{3 \cdot 3 + 2 \cdot 0}{2 + 3} \right) = \left( \frac{-12}{5}, \frac{9}{5} \right)$

Thus, the coordinates of R are R(-\frac{12}{5}, \frac{9}{5}).

Step 2

Find an equation of the line that is perpendicular to L and passes through Q.

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Answer

To find the equation of the line that is perpendicular to L and passes through Q(0, 3), we first need to find the slope of line L (which passes through points P and Q).

The slope (m) of line L is given by:

$m_{L} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{0 - (-6)} = \frac{3}{6} = \frac{1}{2}$

The slope of the line perpendicular to L is given by:

$m_{perpendicular} = -\frac{1}{m_{L}} = -2$

Now, using the point-slope form of the equation of a line, which is:

$y - y_1 = m(x - x_1),$

where (x_1, y_1) are the coordinates of point Q:

Therefore, we have:

$y - 3 = -2(x - 0)$

Simplifying gives:

$y - 3 = -2x$ $y = -2x + 3$

Thus, the equation of the line that is perpendicular to L and passes through Q is:

$y = -2x + 3.$

Here is a sketch of the line L - Edexcel - GCSE Maths - Question 12 - 2021 - Paper 2

Worked Solution & Example Answer:Here is a sketch of the line L - Edexcel - GCSE Maths - Question 12 - 2021 - Paper 2

Find the coordinates of R.

Find an equation of the line that is perpendicular to L and passes through Q.

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