The line segment joining the points A (3, -4) and B (-2, 1) is divided in the ratio 1:3 at point P in it. Find the co-ordinates of P.

Also, find the equation of the line through P and perpendicular to the line 5x - 3y = 4.

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Solution

Using section formula, the co-ordinates of the point P are

$open parentheses fraction numerator 1 cross times negative 2 plus 3 cross times 3 over denominator 1 plus 3 end fraction comma fraction numerator 1 cross times 1 plus 3 cross times negative 4 over denominator 1 plus 3 end fraction close parentheses equals open parentheses 7 over 4 comma fraction numerator negative 11 over denominator 4 end fraction close parentheses equals open parentheses x subscript 1 comma y subscript 1 close parentheses$

The equation of the given line is

5x - 3y + 4 = 0

Slope of this line =

Since, the required line is perpendicular to the given line,

Slope of the required line = $fraction numerator negative 1 over denominator 5 over 3 end fraction equals fraction numerator negative 3 over denominator 5 end fraction$

Thus, the equation of the required line is

y - y1 = m(x - x1)

$y plus 11 over 4 equals fraction numerator negative 3 over denominator 5 end fraction open parentheses x minus 7 over 4 close parentheses fraction numerator 4 y plus 11 over denominator 4 end fraction equals fraction numerator negative 3 over denominator 5 end fraction open parentheses fraction numerator 4 x minus 7 over denominator 4 end fraction close parentheses 20 y plus 55 equals negative 12 x plus 21 12 x plus 20 y plus 34 equals 0 6 x plus 10 y plus 17 equals 0$

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The line segment joining the points A (3, -4) and B (-2, 1) is divided in the ratio 1:3 at point P in it. Find the co-ordinates of P. Also, find the equation of the line through P and perpendicular to the line 5x - 3y = 4.

Using section formula, the co-ordinates of the point P are The equation of the given line is 5x - 3y + 4 = 0 Slope of this line = Since, the required line is perpendicular to the given line, Slope of the required line = Thus, the equation of the required line is y - y1 = m(x - x1)

The line segment joining the points A (3, -4) and B (-2, 1) is divided in the ratio 1:3 at point P in it. Find the co-ordinates of P.

Also, find the equation of the line through P and perpendicular to the line 5x - 3y = 4.