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 Exercise 5.6

If x and y are connected parametrically by the equation given in Exercises 1 to 10 without eliminating the parameter. Find dy/dx.

1. x = 2at2 , y = at4

Solun:- Given x = 2at2 , y = at4

⇒ x = 2at2

Differentiate w.r.t. x:-

{"id":"1","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(2at^{2}\\right)}{x}}\t\n\\end{align*}","ts":1598613043203,"cs":"euxPJWGLI85SgNGBh/f8SQ==","size":{"width":104,"height":36}}

{"type":"align*","id":"2","code":"\\begin{align*}\n{1}&={2a\\diff{\\left(t^{2}\\right)}{x}}\\\\\n{1}&={2a\\times2t\\diff{t}{x}.....\\left(1\\right)}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1598617653496,"cs":"1xmTdKevD7WXnf+9+98Yfw==","size":{"width":158,"height":74}}

⇒ y = at4

Differentiate w.r.t. x:-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"4","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(at^{4}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={4at^{3}\\diff{t}{x}.....\\left(2\\right)}\t\n\\end{align*}","ts":1598617573430,"cs":"FNI/q31qVQSzH1F1P9/CZQ==","size":{"width":154,"height":74}}

Divide 2 by 1:-

{"id":"5","font":{"family":"Arial","color":"#000000","size":12},"type":"$","code":"$\\diff{y}{x}=t^{2}$","ts":1598617910220,"cs":"f6yxEiNhwFGodnJ/Cl4hjQ==","size":{"width":68,"height":26}}

2. x = acos θ , y = bcos θ

Solun:- Given x = acos θ , y = bcos θ

⇒ x = acos θ

Differentiate w.r.t. x:-

{"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(a\\cos\\theta\\right)}{x}}\t\n\\end{align*}","font":{"color":null,"size":10,"family":"Arial"},"id":"6-0","type":"align*","ts":1598618417535,"cs":"kyVPTgTpBx7BB9T9+hV2dQ==","size":{"width":116,"height":33}}

{"type":"$","code":"$1=-a\\sin\\theta\\diff{\\theta}{x}....\\left(1\\right)$","font":{"color":"#000000","family":"Arial","size":10},"id":"7-0","ts":1598618608190,"cs":"V4bEBU3fqtdBkVvyRyZmTg==","size":{"width":149,"height":20}}

⇒ y = bcos θ

Differentiate w.r.t. x:-

{"id":"8-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(b\\cos\\theta\\right)}{x}}\\\\\n{\\diff{y}{x}}&={-b\\sin\\theta\\diff{\\theta}{x}....\\left(2\\right)}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"ts":1598619625041,"cs":"s6hOAZwSUkVEn9SEScFviA==","size":{"width":169,"height":72}}

Divide eq 2 by 1:-

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-b\\sin\\theta}{-a\\sin\\theta}}\\\\\n{\\diff{y}{x}}&={\\frac{b}{a}}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"id":"9-0","type":"align*","ts":1598618883628,"cs":"9BjIlz9FFm9HVi8gwnsG9g==","size":{"width":105,"height":72}}

3. x = sin t , y = cos 2t

Solun:- Given x = sin t , y = cos 2t

⇒ x = sin t

Differentiate w.r.t. x:-

{"type":"align*","font":{"family":"Arial","color":null,"size":10},"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(\\sin t\\right)}{x}}\t\n\\end{align*}","id":"6-1-0","ts":1598619398833,"cs":"Wji+TOLnPlSAJA2HeoYmiQ==","size":{"width":101,"height":33}}

{"code":"$1=\\cos t\\diff{t}{x}....\\left(1\\right)$","id":"7-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","ts":1598619543951,"cs":"kUrP1weWv2eR5tUbBJEMjw==","size":{"width":126,"height":20}}

⇒ y = cos 2t

Differentiate w.r.t. x:-

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"id":"8-1-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(\\cos2t\\right)}{x}}\\\\\n{\\diff{y}{x}}&={-2\\sin2t\\diff{t}{x}....\\left(2\\right)}\t\n\\end{align*}","ts":1598619723020,"cs":"6TEmFNmEisQN9KhEJUvTVw==","size":{"width":177,"height":72}}

Divide eq 2 by 1:-

{"type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-2\\sin 2t}{\\cos t}}\\\\\n{\\diff{y}{x}}&={\\frac{-4\\sin t\\cos t}{\\cos t}}\\\\\n{\\diff{y}{x}}&={-4\\sin t}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"9-1-0","ts":1598619904993,"cs":"lwEnOWwt4rOHdyS3rBeBrA==","size":{"width":136,"height":108}}

4. x = 4t , y = 4/t

Solun:- Given x = 4t , y = 4/t

⇒ x = 4t

Differentiate w.r.t. x:-

{"type":"align*","code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(4t\\right)}{x}}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":null},"id":"6-1-1-0","ts":1598620177691,"cs":"SIuHHyXpJH4J5lL3AXp3pg==","size":{"width":88,"height":33}}

{"type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$1=4.\\diff{t}{x}....\\left(1\\right)$","id":"7-1-1-0","ts":1598620202950,"cs":"OfHmY6Qqe5Yu+XH0q9qzdg==","size":{"width":108,"height":20}}

⇒ y = 4/t

Differentiate w.r.t. x:-

{"font":{"size":10,"color":"#000000","family":"Arial"},"id":"8-1-1-0","type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(\\frac{4}{t}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={-\\frac{4}{t^{2}}\\diff{t}{x}....\\left(2\\right)}\t\n\\end{align*}","ts":1598620347019,"cs":"qNY8Co3dFvJfwVQI6E5hyw==","size":{"width":150,"height":76}}

Divide eq 2 by 1:-

{"id":"9-1-1-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{\\frac{-4}{t^{2}}}{4}}\\\\\n{\\diff{y}{x}}&={\\frac{-1}{t^{2}}}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1598620574474,"cs":"NDG7LWZewrl8mmHipyFohQ==","size":{"width":73,"height":76}}

5. x = cos θ - cos 2θ , y = sin θ - sin 2θ

Solun:- Given x = cos θ - cos 2θ , y = sin θ - sin 2θ

⇒ x = cos θ - cos 2θ

Differentiate w.r.t. x:-

{"type":"align*","id":"6-1-1-1","code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(\\cos\\theta-\\cos2\\theta\\right)}{x}}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":null},"ts":1598621294103,"cs":"eD2xtSValXO7mxq/VYWWkQ==","size":{"width":164,"height":33}}

{"code":"$1=\\left(-\\sin\\theta+2\\sin2\\theta\\right)\\diff{\\theta}{x}....\\left(1\\right)$","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"7-1-1-1-0","ts":1598621624497,"cs":"KoW/a2kwifybLw+qK+/+tQ==","size":{"width":220,"height":20}}

⇒ y = sin θ - sin 2θ

Differentiate w.r.t. x:-

{"type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(\\sin\\theta-\\sin2\\theta\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\left(\\cos\\theta-2\\cos2\\theta\\right)\\diff{\\theta}{x}....\\left(2\\right)}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"8-1-1-1-0","ts":1598621863956,"cs":"rAysEWEP8jZRfzS5BmKXKg==","size":{"width":232,"height":72}}

Divide eq 2 by 1:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{\\cos\\theta-2\\cos2\\theta}{-\\sin\\theta+2\\sin2\\theta}}\\\\\n{\\diff{y}{x}}&={\\frac{\\cos\\theta-2\\cos2\\theta}{2\\sin2\\theta-\\sin\\theta}}\t\n\\end{align*}","id":"9-1-1-1-0","ts":1598622067484,"cs":"Qyf15bpFQOcb1l8UU08qKA==","size":{"width":164,"height":72}}

6. x = a(θ - sin θ) , y = a(1 + cos θ)

Solun:- Given x = a(θ - sin θ) , y = a(1 + cos θ)

⇒ x = a(θ - sin θ)

Differentiate w.r.t. x:-

{"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(a\\left(\\theta-\\sin\\theta\\right)\\right)}{x}}\t\n\\end{align*}","type":"align*","id":"6-1-1-2-0","font":{"size":10,"color":null,"family":"Arial"},"ts":1598689797535,"cs":"Waae//PhLXqe8M9i0cJ0qA==","size":{"width":150,"height":33}}

{"type":"$","code":"$1=a\\left(1-\\cos\\theta\\right)\\diff{\\theta}{x}....\\left(1\\right)$","id":"7-1-1-1-1-0-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1598689830381,"cs":"+56wukddLWmXkAIkQvdHog==","size":{"width":176,"height":20}}

⇒ y = a(1 + cos θ)

Differentiate w.r.t. x:-

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(a\\left(1+\\cos\\theta\\right)\\right)}{x}}\\\\\n{\\diff{y}{x}}&={a\\left(0-\\sin\\theta\\right)\\diff{\\theta}{x}....\\left(2\\right)}\t\n\\end{align*}","id":"8-1-1-1-1-0-0","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1598689969805,"cs":"SMTNjzrVBetgdMxwEkEJQQ==","size":{"width":196,"height":72}}

Divide eq 2 by 1:-

{"id":"9-1-1-1-1-0-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-a\\sin\\theta}{a\\left(1-\\cos\\theta\\right)}}\\\\\n{\\diff{y}{x}}&={\\frac{-2a\\sin\\frac{\\theta}{2}\\cos\\frac{\\theta}{2}}{2a\\sin^{2}\\frac{\\theta}{2}}}\\\\\n{\\diff{y}{x}}&={-\\cot\\frac{\\theta}{2}}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"type":"align*","ts":1598690452511,"cs":"wcvBcAO81xQdivB+cEf4wQ==","size":{"width":154,"height":124}}

{"type":"$","font":{"color":null,"family":"Arial","size":12},"id":"10-0","code":"$7.\\,x=\\frac{\\sin^{3}t}{{\\sqrt[]{\\cos2t}}}\\,,\\,y=\\frac{\\cos ^{3}t}{{\\sqrt[]{\\cos2t}}}$","ts":1598690577801,"cs":"LjPb67JQ/Fw45PHufHrdEw==","size":{"width":221,"height":32}}

Solun:- Given 

{"font":{"color":"#000000","size":12,"family":"Arial"},"id":"11-0-0","code":"$x=\\frac{\\sin^{3}t}{{\\sqrt[]{\\cos2t}}}$","type":"$","ts":1598690627368,"cs":"u3VxU47mddyt0RJCrh1ptQ==","size":{"width":93,"height":32}}

Differentiate w.r.t. x:-

{"id":"6-1-1-3-0","type":"align*","font":{"color":null,"size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(\\frac{\\sin^{3}t}{{\\sqrt[]{\\cos2t}}}\\right)}{x}}\t\n\\end{align*}","ts":1598690650564,"cs":"iwrCOCINfEwPnuupcw6luw==","size":{"width":124,"height":46}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{1}&={\\frac{{\\sqrt[]{\\cos2t}}\\times\\diff{\\left(\\sin^{3}t\\right)}{x}-\\sin^{3}t\\times\\diff{\\left({\\sqrt[]{\\cos2t}}\\right)}{x}}{\\cos2t}\\diff{t}{x}}\\\\\n{1}&={\\frac{3{\\sqrt[]{\\cos2t}}\\times \\sin^{2}t\\times \\cos t-\\sin^{3}t\\times\\frac{\\left(-2\\sin2t\\right)}{2{\\sqrt[]{\\cos2t}}}}{\\cos2t}\\diff{t}{x}}\\\\\n{1}&={\\frac{3\\cos2t\\times \\sin^{2}t\\times \\cos t+\\sin^{3}t\\times\\sin2t}{\\cos2t{\\sqrt[]{\\cos2t}}}\\diff{t}{x}....\\left(1\\right)}\t\n\\end{align*}","type":"align*","id":"7-1-1-1-1-1-0-0","ts":1598692266526,"cs":"c/bDLcLJy0dWFcmQ8qywjA==","size":{"width":368,"height":144}}

{"id":"11-1-0","font":{"family":"Arial","color":"#000000","size":12},"code":"$y=\\frac{\\cos ^{3}t}{{\\sqrt[]{\\cos2t}}}$","type":"$","ts":1598692232265,"cs":"QzmHKSy+YoHnecAAwENMiQ==","size":{"width":92,"height":32}}

Differentiate w.r.t. x:-

{"type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{{\\sqrt[]{\\cos2t}}\\times\\diff{\\left(\\cos ^{3}t\\right)}{x}-\\cos ^{3}t\\times\\diff{\\left({\\sqrt[]{\\cos2t}}\\right)}{x}}{\\cos2t}\\diff{t}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{-3{\\sqrt[]{\\cos2t}}\\times \\cos ^{2}t\\times \\sin t-\\cos ^{3}t\\times\\frac{\\left(-2\\sin2t\\right)}{2{\\sqrt[]{\\cos2t}}}}{\\cos2t}\\diff{t}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{-3\\cos2t\\times \\cos ^{2}t\\times \\sin t+\\cos ^{3}t\\times\\sin2t}{\\cos2t{\\sqrt[]{\\cos2t}}}\\diff{t}{x}....\\left(2\\right)}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"7-1-1-1-1-1-1-0","ts":1598695189053,"cs":"Uo0vHO9pyUjKuwDDMZ6OjQ==","size":{"width":398,"height":142}}

Divide eq 2 by 1:-

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-3\\cos2t\\times \\cos ^{2}t\\times \\sin t+\\cos ^{3}t\\times\\sin2t}{3\\cos2t\\times \\sin^{2}t\\times \\cos t+\\sin^{3}t\\times\\sin2t}}\\\\\n{\\diff{y}{x}}&={\\frac{-3\\cos2t\\times \\cos ^{2}t\\times \\sin t+\\cos ^{3}t\\times2\\sin t\\cos t}{3\\cos2t\\times \\sin^{2}t\\times \\cos t+\\sin^{3}t\\times2\\sin t\\cos t}}\\\\\n{\\diff{y}{x}}&={\\frac{\\sin t\\cos t}{\\sin t\\cos t}\\left(\\frac{-3\\cos2t\\times \\cos t+2\\cos ^{3}t}{3\\cos2t\\times \\sin t+2\\sin^{3}t}\\right)}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"9-1-1-1-1-1-0","type":"align*","ts":1598693452010,"cs":"pCXa8OHJB/rJOIlAikPW8Q==","size":{"width":360,"height":124}}

{"type":"$","code":"$\\diff{y}{x}=\\frac{-3\\cos2t\\times \\cos t+2\\cos ^{3}t}{3\\cos2t\\times \\sin t+2\\sin^{3}t}$","id":"12-0-0","font":{"family":"Arial","size":12,"color":"#000000"},"ts":1598693600142,"cs":"lup7mrBdqRPXFjzy3+Tv9g==","size":{"width":208,"height":29}}

We know that cos 2x = 2cos2x-1

And cos 2x = 1 - 2sin2x

{"type":"align*","id":"12-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-3\\left(2\\cos^{2}t-1\\right)\\times \\cos t+2\\cos ^{3}t}{3\\left(1-2\\sin^{2}t\\right)\\times \\sin t+2\\sin^{3}t}}\\\\\n{\\diff{y}{x}}&={\\frac{-6\\cos^{3}t+3\\cos t+2\\cos ^{3}t}{3\\sin t-6\\sin ^{3}t+2\\sin^{3}t}}\\\\\n{\\diff{y}{x}}&={\\frac{3\\cos t-4\\cos^{3}t}{3\\sin t-4\\sin ^{3}t}}\t\n\\end{align*}","ts":1598694431171,"cs":"aS2MHl0ldqHrQrdkUS5s3A==","size":{"width":276,"height":130}}

We know that cos 3x = 4cos3x - 3cos x

And sin 3x = 3sinx - 4sin3x

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-\\cos3t}{\\sin3t}}\\\\\n{\\diff{y}{x}}&={-\\cot3t}\t\n\\end{align*}","type":"align*","id":"13","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1598694538371,"cs":"bAirE1iQBuakvLmUu8tQ4w==","size":{"width":104,"height":69}}

{"id":"10-1","code":"$8.\\,x=a\\left(\\cos t+\\log_{}\\tan \\frac{t}{2}\\right)\\,,\\,y=a\\sin t$","font":{"family":"Arial","color":null,"size":12},"type":"$","ts":1598694700088,"cs":"CU5WmmhVBWViukVWq9vJLQ==","size":{"width":332,"height":24}}

Solun:- Given 

{"code":"$x=a\\left(\\cos t+\\log_{}\\tan \\frac{t}{2}\\right)$","id":"11-0-1","font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","ts":1598694760431,"cs":"OkOhd765sxtiw5rdGJ72gw==","size":{"width":164,"height":18}}

Differentiate w.r.t. x:-

{"id":"6-1-1-3-1","type":"align*","font":{"size":10,"color":null,"family":"Arial"},"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(a\\left(\\cos t+\\log_{}\\tan \\frac{t}{2}\\right)\\right)}{x}}\t\n\\end{align*}","ts":1598694775935,"cs":"fIB+6jnMCKPc9OmhP8yIwg==","size":{"width":209,"height":36}}

{"code":"\\begin{align*}\n{1}&={a\\left(-\\sin t+\\frac{1}{2\\tan\\frac{t}{2}}\\times \\sec^{2}\\frac{t}{2}\\right)\\diff{t}{x}}\\\\\n{1}&={a\\left(-\\sin t+\\frac{\\cos\\frac{t}{2}}{2\\sin\\frac{t}{2}.\\cos^{2}\\frac{t}{2}}\\right)\\diff{t}{x}}\\\\\n{1}&={a\\left(-\\sin t+\\frac{1}{\\sin t}\\right)\\diff{t}{x}}\\\\\n{1}&={a\\left(\\frac{-\\sin ^{2}t+1}{\\sin t}\\right)\\diff{t}{x}}\\\\\n{1}&={a\\left(\\frac{\\cos^{2}t}{\\sin t}\\right)\\diff{t}{x}}\\\\\n{1}&={a\\left(\\cot t.\\cos t\\right)\\diff{t}{x}....\\left(1\\right)}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","id":"7-1-1-1-1-1-0-1","ts":1598695095209,"cs":"HOg2BsswjfvBpKygxSksjQ==","size":{"width":268,"height":268}}

y=a.sint t

Differentiate w.r.t. x:-

{"font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\diff{y}{x}}&={a\\cos t\\diff{t}{x}.....\\left(2\\right)}\t\n\\end{align*}","id":"7-1-1-1-1-1-1-1","ts":1598695143227,"cs":"kW4NTKQb8ShkZBuLl/aHfQ==","size":{"width":166,"height":32}}

Divide eq 2 by 1:-

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{a\\cos t}{a\\cot t.\\cos t}}\\\\\n{\\diff{y}{x}}&={\\tan t}\t\n\\end{align*}","type":"align*","id":"9-1-1-1-1-1-1","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598695293691,"cs":"tTmGmRzjsUT2JVq4uTTmUA==","size":{"width":129,"height":69}}

9. x = a.sec θ , y = b.tan θ

Solun:- Given x = a.sec θ , y = b.tan θ

⇒x = a.sec θ

Differentiate w.r.t. x:-

{"font":{"color":null,"family":"Arial","size":10},"code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(a\\sec\\theta\\right)}{x}}\t\n\\end{align*}","type":"align*","id":"6-1-1-2-1-0","ts":1598695428638,"cs":"XgXH5smoQ+8qjyz0d9rR+g==","size":{"width":114,"height":33}}

{"type":"$","id":"7-1-1-1-1-0-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"code":"$1=a\\sec\\theta.\\tan\\theta\\diff{\\theta}{x}....\\left(1\\right)$","ts":1598695456956,"cs":"v8YsImyU4Mno0ih9E2zxwQ==","size":{"width":177,"height":20}}

⇒ y = b.tan θ

Differentiate w.r.t. x:-

{"type":"align*","id":"8-1-1-1-1-0-1-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(b\\tan\\theta\\right)}{x}}\\\\\n{\\diff{y}{x}}&={b\\sec^{2}\\theta\\diff{\\theta}{x}....\\left(2\\right)}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1598695742459,"cs":"8hA7ESi8jnqjrovK8sEMSQ==","size":{"width":164,"height":72}}

Divide eq 2 by 1:-

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{b\\sec^{2}\\theta}{a\\sec\\theta .\\tan\\theta}}\\\\\n{\\diff{y}{x}}&={\\frac{b\\sec\\theta}{a\\tan\\theta}}\\\\\n{\\diff{y}{x}}&={\\frac{b}{a}\\left(\\cos ec\\,\\theta\\right)}\t\n\\end{align*}","id":"9-1-1-1-1-0-1-0","type":"align*","ts":1598695904209,"cs":"BTJpWbqCGRYo3rB5ybyf9Q==","size":{"width":133,"height":109}}

10. x = a(cos θ + θsin θ) , y = a(sin θ - θcos θ)

Solun:- Given x = a(cos θ + θsin θ) , y = a(sin θ - θcos θ)

⇒ x = a(cos θ + θsin θ)

Differentiate w.r.t. x:-

{"id":"6-1-1-2-1-1-0","font":{"color":null,"family":"Arial","size":10},"type":"align*","code":"\\begin{align*}\n{\\diff{x}{x}}&={\\diff{\\left(a(cos \\theta + \\theta sin \\theta )\\right)}{x}}\t\n\\end{align*}","ts":1598696096317,"cs":"Vxb9M8yBtAnoV4S9hCiQDA==","size":{"width":180,"height":33}}

{"id":"7-1-1-1-1-0-1-1-0","type":"align*","code":"\\begin{align*}\n{1}&={a\\left(-\\sin\\theta+\\left(\\theta \\cos\\theta+\\sin\\theta\\right)\\right)\\diff{\\theta}{x}}\\\\\n{1}&={a\\left(\\theta \\cos\\theta\\right)\\diff{\\theta}{x}....\\left(1\\right)}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598696188170,"cs":"LHGmiKnD+b8+p065JG+wXg==","size":{"width":240,"height":70}}

⇒ y = a(sin θ - θcos θ)

Differentiate w.r.t. x:-

{"id":"8-1-1-1-1-0-1-1-0","code":"\\begin{align*}\n{\\diff{y}{x}}&={\\diff{\\left(a\\left(\\sin\\theta-\\theta\\cos\\theta\\right)\\right)}{x}}\\\\\n{\\diff{y}{x}}&={a\\left(\\cos\\theta-\\left(-\\theta\\sin\\theta+\\cos\\theta\\right)\\right)\\diff{\\theta}{x}}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":8.703703703703704},"ts":1598696332538,"cs":"TT4JsJhiub+zMY+JMks3qg==","size":{"width":256,"height":72}}

{"type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"14-0","code":"$\\diff{y}{x}=a\\left(\\theta \\sin\\theta\\right)\\diff{\\theta}{x}....\\left(2\\right)$","ts":1598696422708,"cs":"Vuxzu1y/AYFTH/+YhHHUUA==","size":{"width":168,"height":20}}

Divide eq 2 by 1:-

{"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{a\\left(\\theta \\sin\\theta\\right)}{a\\left(\\theta \\cos\\theta\\right)}}\\\\\n{\\diff{y}{x}}&={\\tan\\theta}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"id":"9-1-1-1-1-0-1-1","ts":1598696494571,"cs":"lVVsad6gsnhqLp4UFdPJvQ==","size":{"width":114,"height":76}}

11. If {"type":"$","font":{"family":"Arial","color":"#434343","size":8},"code":"$x={\\sqrt[]{a^{\\sin^{-1}t}}}\\,,\\,y={\\sqrt[]{a^{cos^{-1}t}}}$","id":"15-0","ts":1598697122972,"cs":"2l6ighoIiv8TR6AKA/ycAQ==","size":{"width":153,"height":18}} , show that dy/dx = -y/x

Solun:- Given {"type":"$","font":{"family":"Arial","color":"#434343","size":8},"code":"$x={\\sqrt[]{a^{\\sin^{-1}t}}}\\,,\\,y={\\sqrt[]{a^{cos^{-1}t}}}$","id":"15-0","ts":1598697122972,"cs":"2l6ighoIiv8TR6AKA/ycAQ==","size":{"width":153,"height":18}}

{"id":"16","code":"\\begin{align*}\n{x^{2}}&={a^{\\sin^{-1}t}\\,\\,and\\,\\,y^{2}=a^{cos^{-1}t}}\t\n\\end{align*}","font":{"family":"Arial","size":12,"color":"#000000"},"type":"align*","ts":1598697290523,"cs":"bq5gEje5JFbMzmTWpR1k9Q==","size":{"width":240,"height":25}}

{"id":"17","code":"$x^{2}\\times y^{2}=a^{\\sin^{-1}t+\\cos^{-1}t}$","type":"$","font":{"color":"#000000","family":"Arial","size":12},"ts":1598697338646,"cs":"LwH2hUzLtveBmK6iI7n9WA==","size":{"width":192,"height":24}}

We know that sin-1t + cos-1t = π/2

{"type":"$","font":{"family":"Arial","size":12,"color":"#222222"},"code":"$x^{2}\\times y^{2}=a^{\\frac{\\Pi}{2}}$","id":"18","ts":1598697679775,"cs":"g+4taehrSl4qlv8HLI0Dww==","size":{"width":116,"height":24}}

Differentiate x w.r.t. x:-

{"id":"6-1-1-2-1-1-1","font":{"family":"Arial","size":10,"color":null},"type":"align*","code":"\\begin{align*}\n{x^{2}\\diff{\\left(y^{2}\\right)}{x}+y^{2}\\diff{\\left(x^{2}\\right)}{x}}&={0}\t\n\\end{align*}","ts":1598697744359,"cs":"5qp9/kzfwcaHzgUApx/iYQ==","size":{"width":168,"height":36}}

{"type":"align*","id":"7-1-1-1-1-0-1-1-1","code":"\\begin{align*}\n{2yx^{2}\\diff{y}{x}+2xy^{2}}&={0}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598697874317,"cs":"g/AONw79kCN3+r+uWiZMhA==","size":{"width":134,"height":32}}

{"type":"align*","id":"14-1","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\diff{y}{x}}&={\\frac{-2xy^{2}}{2yx^{2}}}\\\\\n{\\diff{y}{x}}&={\\frac{-y}{x}}\t\n\\end{align*}","ts":1598697955409,"cs":"BcwDbZQ/6CsnSGL2ofxQ6g==","size":{"width":96,"height":76}}

Hence Proved


SEE ALSO:-

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