Straight Lines Part - 1

Questions Discussed:

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

Question 2. The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

Question 3. Find the distance between P (x_1, y_1) \text{ and } Q (x_2, y_2) when :
(i) PQ is parallel to the y-axis,
(ii) PQ is parallel to the x-axis.

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

Question 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).

Question 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.

Questions Discussed:

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

Question 8. Find the value of x for which the points (x, – 1), (2, 1) \text{ and } (4, 5) are collinear.

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

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

Question 11. The slope of a line is double of the slope of another line. If tangent of the angle between them is { \displaystyle \frac{1}{3} }, find the slopes of the lines.

Question 12. A line passes through { \displaystyle(x_1, y_1) } \text{ and } (h, k). If slope of the line is m, show that { \displaystyle k – y_1 = m (h – x_1) }.

Question 13. If three points (h, 0), (a, b) \text{ and } (0, k) lie on a line, show that { \displaystyle \frac{a}{h} } + { \displaystyle \frac{b}{k} } = 1.

Question 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?

Questions Discussed:

In Exercises 1 to 8, find the equation of the line which satisfy the given conditions:

Question 1. Write the equations for the x-and y-axes.

Question 2. Passing through the point (– 4, 3) with slope { \displaystyle \frac{1}{2} }

Question 3. Passing through (0, 0) with slope m.

Question 4.Passing through ( 2, 2\sqrt{3} ) and inclined with the x-axis at an angle of 75^\circ.

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

Questions Discussed:

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

Question 6. Intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30^\circ with positive direction of the x-axis.

Question 7. Passing through the points (–1, 1) and (2, – 4).

Question 8. Perpendicular distance from the origin is 5 units and the angle made by the perpendicular with the positive x-axis is 30^\circ.

Questions Discussed:

Question 8. Perpendicular distance from the origin is 5 units and the angle made by the perpendicular with the positive x-axis is 30^\circ.

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

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

Questions Discussed:

Question 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.

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

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

Question 14. Find equation of the line through the point (0, 2) making an angle { \displaystyle \frac{2\pi}{3} } 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.

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

Questions Discussed:

Question 16. The length L (in centimetre) 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.

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

Question 18. P(a, b) is the mid-point of a line segment between axes. Show that equation of the line is { \displaystyle \frac{x}{a} + \frac{y}{b} = 2}

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

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