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Explanation of formula of class 11 mathematics chapter 10 straight line distance formula, section formula , midpoint formula, area of triangle when 3 coordinate points are given

**Question 1:** **Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.**

Answer

Draw a Quadrilateral ABCD whose vertices are A (–4, 5), B (0, 7), C (5, –5), and D (–4, –2).

Area of quadrilateral ABCD = area of ∆ABC + area of ∆ACD

Relate the coordinate of ∆ ABC, A (–4, 5), B (0, 7), C (5, –5) with (x1, y1), (x2, y2), and (x3, y3) and use formula

\[ = \frac{1}{2}|{x_1}({y_2} - {y_3}+{x_2}({y_3} –{y_1} )+ +{x_3}({y_1} –{y_2} |\]

Plug the values, we get

=1/2|-4(7+5) + 0(-5 ,-5) + 5(5-7)|

=1/2|-4(12 +0 + 5(-2)|

=1/2|-48 -10|

=1/2 (58)

=29 unit^{2}

Relate the coordinate of ∆ ABC, A (–4, 5), C (5, –5) and D (–4, –2). with (x1, y1), (x2, y2), and (x3,y3) and use formula

\[ = \frac{1}{2}|{x_1}({y_2} - {y_3}+{x_2}({y_3} –{y_1} )+ +{x_3}({y_1} –{y_2} |\]

Area of ∆ACD

=1/2|-4(-5+ 2) +5(-2-5) + (-4)(5 +5)|

= ½ |-4(-3) +5(-7) +(-4)(10)|

=1/2|12 -35 -40|

=1/2|-63|

= 63/2 unit^{2 }

Total area of quadrilateral (ABCD) = 29+ 63/2 = 121/2 unit^{2}

**3. Find the distance between P (x1, y1
and Q (x2, y2) when : (i) PQ is parallel to they-axis, (ii) PQ is parallel to the x-axis.
**

**4. Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
**

**5. Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0).**

**6. Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and
(–1, –1) are the vertices of a right angled triangle.**

**7. Find the slope of the line, which makes an angle of 30° with the positive direction
of y-axis measured anticlockwise.**

**8. Find the value of x for which the points (x, – 1), (2,1) and (4, 5) are collinear.
**

**9. Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram.
**

**10. Find the angle between the x-axis and the line joining the points (3,–1) and (4,–2).
**

**11. The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines.
**

**12. A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).**

**13. If three points (h, 0), (a, b) and (0, k) lie on a line, show that
a/h + b/k = 0**

**14. Consider the following population and year graph (Fig 10.10), find the slope of the line AB and using it, find what will be the population in the year 2010?**

**Question 1-3 (1)Write the equations for the x-and y-axes. (2) Passing through the point (– 4, 3) with slope 1/2
(3) Passing through (0, 0) with slope m.**

**5. Intersecting the x-axis at a distance of 3 units to the left of origin with slope –2.**

**7. Find equation of line passing Passing through the points (–1, 1) and (2, – 4).
**

**8. Find the equation of line whose Perpendicular distance from the origin is 5 units and the angle made by the
perpendicular with the positive x-axis is 30.
**

**9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.
**

**10. Find the equation of the line passing through (–3, 5) and perpendicular to the line
through the points (2, 5) and (–3, 6).
**

**11. A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides
it in the ratio 1: n. Find the equation of the line.
**

**12. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
**

**13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.**

**14. Find equation of the line through the point (0, 2) making an angle 2π/3
with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.
**

**15. The perpendicular from the origin to a line meets it at the point (–2, 9), find the equation of the line.
**

**16. The length L (in centimetrs) of a copper rod is a linear function of its Celsius
temperature C. In an experiment, if L = 124.942 when C = 20 and L= 125.134
when C = 110, express L in terms of C.**

**17. The owner of a milk store finds that, he can sell 980 litres of milk each week at
Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?**

**18. P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is x/a + y/b = 2**

**19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find equation of the line.
**

**20. By using the concept of equation of a line, prove that the three points (3, 0),(– 2, – 2) and (8, 2) are collinear.**

**1. Reduce the following equations into slope - intercept form and find their slopes
and the y - intercepts.
(i) x + 7y = 0, (ii) 6x + 3y – 5 = 0, (iii) y = 0.
**

**2. Reduce the following equations into intercept form and find their intercepts on the axes.
(i) 3x + 2y – 12 = 0, (ii) 4x – 3y = 6, (iii) 3y + 2 = 0.**

**3. Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
(i) x –3y + 8 = 0, (ii) y – 2 = 0, (iii) x – y = 4.
**

**4. Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).**

**5. Find the points on the x-axis, whose distances from the line x/3 + y/4 =1
are 4 units.**

**6. Find the distance between parallel lines(i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0 (ii) l (x + y) + p = 0 and l (x + y) – r = 0.
**

**7. Find equation of the line parallel to the line 3x- 4y + 2 = 0 and passing through the point (–2, 3).**

**8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.**

**9. Find angles between the lines
**

**10. The line through the points (h, 3) and (4, 1) intersects the line 7x -9y -19 = 0
at right angle. Find the value of h.
**

**11. Prove that the line through the point (x 1 , y1 ) and parallel to the line Ax + By + C = 0 is A (x –x 1 ) + B (y – y1 ) = 0.**

**12. Two lines passing through the point (2, 3) intersects each other at an angle of 60 If slope of one line is 2, find equation of the other line.
**

**14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.**

**15. The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.**

**16. If p and q are the lengths of perpendiculars from the origin to the
lines... **

**17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation
and length of altitude from the vertex A.
**

**18. If p is the length of perpendicular from the origin to the line whose intercepts on ...**

**1. Find the values of k for which the line (k–3) x – (4 – k2) y + k2 –7k + 6 = 0 is (a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.
**

**2. Find the values of θ and p, if the equation x cos θ + y sinθ = p is the normal form ...**

**3. Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and – 6, respectively.**

**4. What are the points on the y-axis whose distance from the line
is ... 4 units.**

**5. Find perpendicular distance from the origin of the line joining the points (cosθ, sin θ)
and (cos φ, sin φ).**

**6. Find the equation of the line parallel to y-axis and drawn through the point ofintersection of the lines x – 7y + 5 = 0 and 3x + y = 0.**

**7. Find the equation of a line drawn perpendicular to the line ... through the point, where it meets the y-axis.**

**8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0.
**

**9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.
**

**10. If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m
are concurrent, then show that ...**

**11. Find the equation of the lines through the point (3, 2) which make an angle of 45 with the line x – 2y = 3.
**

**12. Find the equation of the line passing through the point of intersection of the lines
4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes**

**13. Show that the equation of the line passing through the origin and making an angle θ with the line ...**

**14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
**

**15. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
**

**16. Find the direction in which a straight line must be drawn through the point (–1, 2)
so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.**

**17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find the equation of the legs (perpendicular sides) of the triangle.**

**18. Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the
line to be a plane mirror.**

**23. Prove that the product of the lengths of the perpendiculars drawn from the ...**

**24. A person standing at the junction (crossing) of two straight paths represented by
the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose
equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he
should follow.**

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