{"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":"

Solution :<\/h2>\n

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

\n

Similar Questions<\/h3>\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

If PS is the median of the triangle, with vertices of P(2,2), Q(6,-1) and R(7,3), then equation of the line passing through (1,-1) and parallel to PS is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : The given line is bx + ay – ab = 0 ………….(i) It is given that p = Length of the perpendicular from the origin to line (i) \\(\\implies\\) p = \\(|b(0) + a(0) – ab|\\over {\\sqrt{b^2+a^2}}\\) = \\(ab\\over \\sqrt{a^2+b^2}\\) \\(\\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>\n

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\\)<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,48],"tags":[],"yoast_head":"\nIf p is the length of the perpendicular from the origin to the line \\(x\\over a\\) + 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sqrt{a^2+b^2}) (implies) (p^2) = (a^2b^2over a^2+b^2) (implies) (1over p^2) = (a^2+b^2over a^2b^2) (implies) (1over … If p is the length of the perpendicular from the origin to the line (xover a) + (yover b) = 1, then prove that (1over p^2) = (1over a^2) + (1over b^2) Read More »\" \/>\n<meta property=\"og:url\" content=\"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\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-22T10:30:58+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-25T04:12:07+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. 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