List of practice Questions

Find the area of the triangle formed by the lines  \(y -x = 0, x + y = 0\) and \(x - k = 0.\)

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that  \(\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}.\)

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

What are the points on the y-axis whose distance from the line  \(\frac{ x}{3} +\frac{ y}{4} = 1\) is 4 units?

Find the equation of the right bisector of the line segment joining the points (3,4) and (–1,2).

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

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.

If p and q are the lengths of perpendiculars from the origin to the lines  \(x \space cos θ − y \space sin θ = k\space  cos 2θ\) and \(x \space sec θ + y\space cosec θ = k\), respectively, prove that  \(p^2 + 4q^2 = k^2\)

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.

Prove that the line through the point \((x_1, y_1)\) and parallel to the line \(Ax + By + C = 0 \) is \(A (x -x_1) + B (y - y_1) = 0.\)

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

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

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

Find angles between the lines \(\sqrt3x+y=1\) and \(x+\sqrt3y=1.\)

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

Find the points on the x-axis, whose distances from the line \(\frac{ x}{3}+\frac{y}{4}=1\) are 4 units.

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

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) \(y = 0\)

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

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

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

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?

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

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.

Prove that the product of the lengths of the perpendiculars drawn from the points \(\left(\sqrt{a^2-b^2},0\right)\) and \(\left(-\sqrt{a^2-b^2},0\right)\) to the line \(\frac{x}{a}cosθ+\frac{y}{b}sinθ=1\) is \(b^2\).