Subjective Questions
Q1:
Find the co-ordinates of the mid point of the line segement joining the given points, (4, 6) and (2, 4).
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Here,<br>(4, 6) = (x<sub>1</sub>, y<sub>1</sub>)<br>(2, 4) = (x<sub>2</sub>, y<sub>2</sub>)<br>The co-ordinates of the mid point = (\(\frac{x_1 + x_2}{2}\), \(\frac{y_1 + y_2}{2}\))<br>= (\(\frac{4+ 2}{2}\), \(\frac{6 + 4}{2}\))<br>= (\(\frac{6}{2}\), \(\frac{10}{2}\))<br>= (3, 5)</p> <p>\(\therefore\) The co-ordinates of given mid point is (3, 5).</p>
Q2:
Find the co-ordinates of the point (2, 3) and (7, 3) which divides the line segment joining the following points in the 3:2 ratio.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Here,<br>(2, 3) = (x<sub>1</sub>, y<sub>1</sub>)<br>(7, 3) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> : m<sub>2</sub> = 3 : 2<br>Now,<br>or, (x, y) = (\(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\), \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\))<br>or, (x, y) = (\(\frac{3\times7 + 2\times2}{3 + 2}\), \(\frac{3\times3 + 2\times3}{3 + 2}\))<br>or, (x, y) = (\(\frac{21 + 4}{5}\), \(\frac{9 + 6}{5}\))<br>or, (x, y) = (\(\frac{25}{5}\), \(\frac{15}{5}\))<br>or, (x, y) = (5, 3)<br>So, the required point is (5, 3).</p>
Q3:
The mid point of a line is (0, 3) and one end of the line is (−6, 8). Find the co-ordinates of the other end.
Type: Long
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(0, 3) = (x, y)<br>(-6, 8) = (x<sub>1</sub>, y<sub>1</sub>)<br>Let co-ordinates of other end be (x<sub>2</sub>, y<sub>2</sub>)<br>Now,<br>or, x = \(\frac{x_1 + x_2}{2}\)<br>or, 0 = \(\frac{-6 + x_2}{2}\)<br>or, 0 = - 6 + x<sub>2</sub><br>\(\therefore\) x<sub>2 = </sub>6<sub><br></sub>Again,<br>or, y = \(\frac{y_1 + y_2}{2}\)<br>or, 3 = \(\frac{8 + y_2}{2}\)<br>or, 6 = 8 + y<sub>2</sub><br>or, 6 - 8 = y<sub>2</sub><br>\(\therefore\) y<sub>2</sub>= -2</p> <p>So, the co-ordinates of other end is (6, -2).</p>
Q4:
Find the ratio the line joining the points (2, 2) and (−2, −2) is divided by the point (\(\frac{6}{7}\), \(\frac{6}{7}\)) ?
Type: Long
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Given,<br>(2, 2) = (x<sub>1</sub>, y<sub>1</sub>)<br>(-2,-2) = (x<sub>2</sub>, y<sub>2</sub>)<br>(x, y) = (\(\frac{6}{7}\), \(\frac{6}{7}\))<br>m<sub>1</sub> : m<sub>2</sub> = ?<br>By using section formula,<br>or, x = \(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\)<br>or, \(\frac{6}{7}\) =\(\frac{m_1 \times(- 2) + m_2 \times 2}{m_1 + m_2}\)<br>or, 6(m<sub>1</sub>+ m<sub>2</sub>) = 7(-2m<sub>1</sub> + 2m<sub>2</sub>)<br>or, 6m<sub>1</sub>+ 6m<sub>2</sub> = -14m<sub>1</sub> + 14m<sub>2<br></sub>or, 6m<sub>1</sub>+ 14m<sub>1</sub>=14m<sub>2 </sub>-6m<sub>2</sub><br>or, 20m<sub>1</sub> = 8m<sub>2</sub><br>or, \(\frac{m_1}{m_2}\) = \(\frac{8}{20}\)<br>\(\therefore\) m<sub>1</sub> : m<sub>2</sub>= 2 : 5<br>Now,<br>or, y = \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\)<br>or, \(\frac{6}{7}\) = \(\frac{2\times (-2)+ 5\times2}{2 + 5}\)<br>or, \(\frac{6}{7}\) = \(\frac{(-4)+ 10}{7}\)<br>or, \(\frac{6}{7}\) = \(\frac{6}{7}\) <br>\(\therefore\) The required ratio is 2 : 5.</p>
Q5:
Find the ratio in which the y-axis divides the line segment joining the points (−4, 1) and (10, 1)?
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Let the points on y-axis be (0, y) which divides the line joining the points<br>(-4, 1) = (x<sub>1</sub>, y<sub>1</sub>) and (10, 1) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> : m<sub>2</sub> = ?<br>By using sectional formula<br>or, x = \(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\)<br>or, 0 =\(\frac{m_1 \times10+ m_2 \times(-4)}{m_1 + m_2}\)<br>or, 0 = 10m<sub>1 </sub>- 4m<sub>2</sub><br>or, \(\frac{m_1}{m_2}\) = \(\frac{4}{10}\)<br>or, \(\frac{m_1}{m_2}\) = \(\frac{2}{5}\)<br>\(\therefore\) m<sub>1</sub> : m<sub>2</sub> = 2 : 5</p>
Q6:
Find the ratio in which x-axis divides the line segment joining the points (3, 2) and (3, −9)?
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Let the points on y-axis be (x, 0) which divides the line joining the point<br> (3, 2) = (x<sub>1</sub>, y<sub>1</sub>)<br>(3,-9) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> : m<sub>2</sub>= ?<br>By using section formula,<br>or, y = \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\)<br>or, 0 =\(\frac{m_1\times(-9)+ m_2 \times2}{m_1 + m_2}\)<br>or, 0 = -9m<sub>1</sub>+ 2m<sub>2</sub><br>or, \(\frac{m_1}{m_2}\) = \(\frac{2}{9}\)<br>\(\therefore\)m<sub>1</sub> : m<sub>2</sub>= 2 : 9<br><br></p>
Q7:
Find the co-ordinates of the mid point of the line segment joining the given points (5a, 7b) and (3a, −2b).
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Given,<br>(5a, 7b) = (x<sub>1</sub>, y<sub>1</sub>)<br>(3a,-2b) = (x<sub>2</sub>, y<sub>2</sub>)<br>Co-ordinates of mid point = (\(\frac{x_1 + x_2}{2}\), \(\frac{y_1 + y_2}{2}\))<br>= (\(\frac{5a + 3a}{2}\), \(\frac{7b- 2b}{2}\))<br>= (\(\frac{8a}{2}\), \(\frac{5b}{2}\))<br>= (4a, \(\frac{5b}{2}\))</p>
Q8:
Find the co-ordinates of the point which divides the line segment joining the (2, −4) and (−3, 6) points in the 2:3 ratio.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(2,-4) = (x<sub>1</sub>, y<sub>1</sub>)<br>(-3, 6) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> : m<sub>2</sub> = 2 : 3<br>Now,<br>or, (x, y) = (\(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\), \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\))<br>or, (x, y) = (\(\frac{2\times (-3) + 3\times2}{2 +3}\), \(\frac{2\times6 + 3\times (-4)}{2 + 3}\))<br>or,(x, y) = (\(\frac{- 6+ 6}{5}\), \(\frac{12-12}{5}\))<br>or, (x, y) = (0, 0)<br>\(\therefore\)The required point is (0, 0).<br><br></p>
Q9:
The mid ppoint of a line is (−4, −5) and one end of the line is (−5, −3). Find the co-ordinates of the other end.
Type: Long
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(-4, -5) = (x, y)<br> (-5, -3) = (x<sub>1</sub>, y<sub>1</sub>)<br>Let the co-ordinates of the other end be (x<sub>2</sub>, y<sub>2</sub>)<br>or, x = \(\frac{x_1 + x_2}{2}\)<br>or, -4 = \(\frac{-5 + x_2}{2}\)<br>or, -8 = -5 + x<sub>2</sub><br>or, -8 + 5 = x<sub>2</sub><br>or, x<sub>2</sub> = -3<br>Again, <br>or, y = \(\frac{y_1 + y_2}{2}\)<br>or, -5 = \(\frac{-3 + y_2}{2}\)<br>or, -10 = -3 + y<sub>2</sub><br>or, y<sub>2</sub> = -10 + 3<br>or, y<sub>2</sub> = -7</p> <p>\(\therefore\) The required point(x<sub>2</sub>, y<sub>2</sub>) = (-3, -7)</p> <p></p>
Q10:
In what ratio the line joining the points (2, -4) and (-3, 6) is divided by the point (0, 0) ?
Type: Long
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(2, -4) = (x1, y1)<br>(-3, 6) = (x2, y2)<br>(0, 0) = (x, y)<br>By using section formula<br>or, x = \(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\)<br>or, 0 = \(\frac{m_1 \times\ (-3) + m_2 \times 2}{m_1 + m_2}\)<br>or, 0 = \(\frac{-3m_1 + 2m_2}{m_1 + m_2}\)<br>or, 0 = -3m<sub>1</sub> + 2m<sub>2</sub><br>or, 3m<sub>1</sub> = 2m<sub>2</sub><br>or, \(\frac{m_1}{m_2}\) = \(\frac{2}{3}\)<br>\(\therefore\) m<sub>1</sub> : m<sub>2</sub> = 2 : 3<br>Now,<br>or, y = \(\frac{m_1 y_2 + m_2 y_1}{2 + 3}\)<br>or, 0 = \(\frac{2 \times\ 6 + 2\times\ (-4)}{m_1 + m_2}\)<br>or, 0 = \(\frac{12 - 12}{5}\)<br>or, 0 = 0<br>\(\therefore\) The required ratio is 2 : 3</p>
Q11:
Find the co-ordinates of the point which divides the line segment joining (-10, 12) and (-3, -9) points in the 4:3 ratio.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(-10, 12) = (x<sub>1</sub>, y<sub>1</sub>)<br>(-3, -9) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> ; m<sub>2</sub> = 4 : 3<br>Now,<br>or, (x, y) = (\(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\), \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\))<br>or, (x, y) = (\(\frac{4 \times(-3) + 3 \times(-10)}{4 + 3}\), \(\frac{4 \times(-9) + 3 \times12}{4 + 3}\))<br>or, (x, y) = (\(\frac{-12 - 30}{7}\), \(\frac{-36 + 36}{7}\))<br>or, (x, y) = (-6, 0)<br>\(\therefore\) The required points are (-6, 0)</p>
Q12:
Find the co-ordinates of the mid point of the line segment joining the (8, 5) and (-12, -7) points.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(8, 5) = (x<sub>1</sub>, y<sub>1</sub>)<br>(-12, -7) = (x<sub>2</sub>, y<sub>2</sub>)</p> <p>Co-ordinates of the mid point = (\(\frac{x_1 + x_2}{2}\), \(\frac{x_1 + y_2}{2}\))<br>= (\(\frac{8 - 12}{2}\), \(\frac{5 - 7}{2}\))<br>= (\(\frac{- 4}{2}\), \(\frac{- 2}{2}\))<br>= (-2, -1)</p>
Q13:
If the ratio of line joining points (-1, 3) and (4, 8) is divided by the point (2 6) ?
Type: Long
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(-1, 3) = (x<sub>1</sub>, y<sub>1</sub>)<br> (4, 8) = (x<sub>2</sub>, y<sub>2</sub>)<br>(2, 6) = (x, y)<br>By using section formula<br>or, x = \(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\)<br>or, 2 = \(\frac{m_1 \times\ 4 + m_2 \times\ (-1)}{m_1 + m_2}\)<br>or, 2(m<sub>1</sub> + m<sub>2</sub>) = 4m<sub>1</sub> - 1m<sub>2</sub><br>or, 2m<sub>1</sub> + 2m<sub>2</sub> = 4m<sub>1</sub> - 1m<sub>2</sub><br>or, 2m<sub>1</sub>- 4m<sub>1</sub> = -1m<sub>2</sub> - 2m<sub>2</sub><br>or, -2m<sub>1</sub> = -3m<sub>2</sub><br>Now,<br>or, y = \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\)<br>or, 6 = \(\frac{-3\times8 + (-2)\times\ 3}{-3 + (-2)}\)<br>or, 6 = \(\frac{-24 -6}{-3 - 2}\)<br>or, 6 = \(\frac{-30}{-5}\)<br>or, -30 = -30</p> <p>\(\therefore\) The required ratio is (-3, -2)</p>
Q14:
Find the co-ordinates of the mid point of the line segment joining (-3, -7) and (-5, -3) points.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>Given,<br>(-3, -7) = (x<sub>1</sub>, y<sub>1</sub>)<br> (-5, -3) = (x<sub>2</sub>, y<sub>2</sub>)<br>Co-ordinates of the mid point = (\(\frac{x_1 + x_2}{2}\), \(\frac{y_1 + y_2}{2}\))<br>= (\(\frac{-3 + (-5)}{2}\), \(\frac{-7 + (-3)}{2}\))<br>= (\(\frac{-3 - 5}{2}\), \(\frac{-7 - 3}{2}\))<br>= (\(\frac{-8}{2}\), \(\frac{-10}{2}\))<br>= (-4, -5)</p>
Q15:
Find the co-ordinates of the point which divides the line segment joining (2, 2) and (-2, -2) points in the 2:5 ratios.
Type: Short
Difficulty: Easy
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Answer: <p>Solution:</p> <p>(2, 2) = (x<sub>1</sub>, y<sub>1</sub>)<br>(-2, -2) = (x<sub>2</sub>, y<sub>2</sub>)<br>m<sub>1</sub> : m<sub>2</sub> = 2 : 5<br>now,<br>or, (x, y) = (\(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\), \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\))<br>or, (x, y) = (\(\frac{2 \times\ (-2) + 5 \times\ 2}{2 + 5}\), \(\frac{2 \times\ (-2) + 5 \times\ 2}{2 + 5}\))<br>or, (x, y) = (\(\frac{-4 + 10}{7}\), \(\frac{-4 + 10}{7}\))<br>or, (x, y) = (\(\frac{6}{7}\), \(\frac{6}{7}\))</p> <p>\(\therefore\) The required points is (\(\frac{6}{7}\), \(\frac{6}{7}\))</p>
