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- Different forms of Ax + By + C = 0 - Slope-intercept form, Intercept form, Normal form
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and xsec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2
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.
Find equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
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.
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 – sqrt3y + 8 = 0`
(ii) y – 2 = 0
(iii) x – y = 4
Prove that the product of the lengths of the perpendiculars drawn from the points
`(sqrt(a^2 - b^2),0)` and `(-sqrta^2-b^2,0)` to the line `x/a cos theta + y/b sin theta = 1` is `b^2`
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.
Show that the equation of the line passing through the origin and making an angle θ with the line `y = mx + c " is " y/c = (m+- tan theta)/(1 +- m tan theta)`
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.
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.
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
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.
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x –x1) + B (y – y1) = 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 is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that `1/p^2 = 1/a^2 + 1/b^2`
If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.