{"id":7058,"date":"2021-10-22T16:00:58","date_gmt":"2021-10-22T10:30:58","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7058"},"modified":"2021-10-25T09:42:07","modified_gmt":"2021-10-25T04:12:07","slug":"if-p-is-the-length-of-the-perpendicular-from-the-origin-to-the-line-xover-a-yover-b-1-then-prove-that-1over-p2-1over-a2-1over-b2","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/if-p-is-the-length-of-the-perpendicular-from-the-origin-to-the-line-xover-a-yover-b-1-then-prove-that-1over-p2-1over-a2-1over-b2\/","title":{"rendered":"If p is the length of the perpendicular from the origin to the line \\(x\\over a\\) + \\(y\\over b\\) = 1, then prove that \\(1\\over p^2\\) = \\(1\\over a^2\\) + \\(1\\over b^2\\)"},"content":{"rendered":"
The given line is bx + ay – ab = 0 ………….(i)<\/p>\n
It is given that<\/p>\n
p = Length of the perpendicular from the origin to line (i)<\/p>\n
\\(\\implies\\) p = \\(|b(0) + a(0) – ab|\\over {\\sqrt{b^2+a^2}}\\) = \\(ab\\over \\sqrt{a^2+b^2}\\)<\/p>\n
\\(\\implies\\) \\(p^2\\) = \\(a^2b^2\\over a^2+b^2\\) \\(\\implies\\) \\(1\\over p^2\\) = \\(a^2+b^2\\over a^2b^2\\) \\(\\implies\\) \\(1\\over p^2\\) = \\(1\\over a^2\\) + \\(1\\over b^2\\)<\/p>\n
Hence Proved.<\/p>\n
Find the distance between the line 12x \u2013 5y + 9 = 0 and the point (2,1)<\/a><\/p>\n If the line 2x + y = k passes through the point which divides the line segment joining the points (1,1) and (2,4) in the ratio 3:2, then k is equal to<\/a><\/p>\n The x-coordinate of the incenter of the triangle that has the coordinates of mid-point of its sides as (0,1), (1,1) and (1,0) is<\/a><\/p>\n Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax+2ay+c=0 and 5bx+2by+d=0 lies in the fourth quadrant and is equidistant from the two axes, then<\/a><\/p>\n