Maths-
General
Easy
Question
The normal to the curve x = 3 cos
–
, y = 3 sin
– ![sin cubed](data:image/png;base64,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)
at the point
=
/4 passes through the point -
- (2, –2)
- (0, 0)
- (–1, 1)
- None of these
Hint:
Find, Slope of tangent =
= ![fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADoAAAAxCAYAAACLfLrrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAeOiuv/QAAAulJREFUeNrtmU+EVVEcx6+h0Z8RNcbQ7DKLFhlD8iSVSMbzekabjEqeaFFkdikyi0S1aJEoMpIkxmiRJDKSSpskSYuYNk9tIpnFSJvX96fv5Tjuufe8e+/M/fWcLx/evefd455/v3Pu7xtF1WtdpFzLoK9gHVtAW3NjR8HHEuq5Da5oHs1J8LCEzvoBNmlp1AC4AX6B32Ae3ATnWD4LjiQ8Nw0up9QrHXVeSyNlDb4ALf7eCGZAh6MqOgGuWc+tB1/AoKPecfCN/1Ohi8bImZJGjPB3HcxZ5TN81qWn4LSmtdjm1LVHedm4llH7ZFwPgUWw1lHnXpavKRjhd4NHnBkvQS1vI8dZga0ap7P9IrGuc3269AYcLRjha2xkn9W5uXQY3E+4PwXuWvdega1kMWV/PcSX7mb/TYrwH4ylE+sZGMu7hcwl3JfGn7Lu3QNN8AAcTwls0shGgQgvOpDQ0XEUr+dpqETEz5zCoh3gCddEPWErmc04RBwDrwtGeNEdzjZbMpUP5p2+w6xAevct2A9+JgSaSb5Q01GPBJ6vYE/BCB+x8zsOhlc6OjfZES6d4ZZSNMLL9ZJjNiytxjYkjdjlKNsAvhtLoEiElxFbSPjfTq7nFdV2rl2XLnici30jfLytRAkHlFaVB47NPLiPlhThB7i1mJKPgvegv8qGXgW3So7wcgo7y9/bGBsaVWcO2vy4LjPCj3FU/3A7a2o4K6tPkwQFBelTp0cJCgoK+v9PUGUYTLHUGk1lGUyx1BpNZRhMZqepMJp80o8nHSNyiWVpUmE0+aYfI37hmxm4FjskTWqMJt/0o2gi+mdFiCTBvOBRvxqjySf9aOo52Mev/8GMutUYTd0YTLGmmd7w8T/UGE3dGExxz85z+k5l1K3KaOrGYBIH7TGDikz1dylTV53R5Jt+HGJQMRvWcMwGkUqjKSv92M/yEUfPTlj3esZoylLPGE1p6hmjKUvBaMoR6YPRVHXmIBhNq6m/l2UoLP+SwzIAAAFmdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtc3R5bGUgZGlzcGxheXN0eWxlPSJ0cnVlIj48bWZyYWMgYmV2ZWxsZWQ9InRydWUiPjxtcm93PjxtaT5kPC9taT48bWk+eTwvbWk+PC9tcm93Pjxtcm93PjxtaT5kPC9taT48bWk+JiN4M0I4OzwvbWk+PC9tcm93PjwvbWZyYWM+PC9tc3R5bGU+PG1zdHlsZSBkaXNwbGF5c3R5bGU9InRydWUiPjxtZnJhYyBiZXZlbGxlZD0idHJ1ZSI+PG1yb3c+PG1pPmQ8L21pPjxtaT54PC9taT48L21yb3c+PG1yb3c+PG1pPmQ8L21pPjxtaT4mI3gzQjg7PC9taT48L21yb3c+PC9tZnJhYz48L21zdHlsZT48L21mcmFjPjwvbWF0aD74luT2AAAAAElFTkSuQmCC)
Then find slope of normal, and substitute the values in equation of normal line.
The correct answer is: (0, 0)
Given :
x = ![3 cos theta space minus space cos cubed theta](data:image/png;base64,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)
![y space equals space 2 sin theta space minus space sin cubed theta](data:image/png;base64,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)
At ![theta space equals space straight pi over 4](data:image/png;base64,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)
![x space equals space 3 cos straight pi over 4 space minus space space cos cubed straight pi over 4 space equals space 3 cross times fraction numerator 1 over denominator square root of 2 end fraction space minus space fraction numerator 1 over denominator 2 square root of 2 end fraction space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction
y space equals space 2 sin straight pi over 4 space minus space sin cubed straight pi over 4 space equals space 2 cross times fraction numerator 1 over denominator square root of 2 end fraction space minus space fraction numerator 1 over denominator 2 square root of 2 end fraction space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction](data:image/png;base64,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)
At P(
= ![left parenthesis fraction numerator 5 over denominator 2 square root of 2 end fraction comma space fraction numerator 5 over denominator 2 square root of 2 end fraction right parenthesis](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGsAAAAqCAYAAAC0u6XFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAotJREFUeNrtmz9IHEEUxgcREbGIELCIEJErJAQRVFCStCJWIgqKbQgiIpIqhQE7xSZFygTSRRD/pAgaCEHkkKCNElIEg1haCAoWJhz++573OI9jb2/nbsebGd8HP+7YvXu73zx2dt7srFJmdRWCq/LRU8aYb/LRkyRLjIknMSbJShtLgVOwCl6DWvFktypBG3gL/oBm8eSGusGGeHJH/8VT6RoDs4aP8RTse5YoE55mOR+BagE/Yz7gMugCFUwPm+p3ODF36WmT8xK4ozXmgw2CPXAOjsEC6HD8KrpLTzSASeZubOVkiexTkpOW0RwYl3axUhO544hv4IW0i5Wi++OP7A1UjVdJu1ipKs5PRufSJlYrFSVZV5ZTjFz0lJJu0NFu8Dt4Ju3ixgBDhu72apzz41VRPM99fp/vRTGJ5gVbJFlWKXC6SSkzE7n5VGtoFEgJ6y1Tw5rwtBl2AY0q849ISENgPeaY1JX/VuknuOVQ3J7oPvXKhst7DbyMOeYjUO+Zp7LrITjhT/FkSFSHLXKBRhX1LhgpIg51tSsG4993TzeiBSHD6nbZ1RO+AQ5rxqF+fcBg/PvuKa8eg18av28ER6DaUHzxVEA6q3regA8G44unEHVpznzsqPQCE1PxxVMe0WW/pYIngIOWE9NyrUONOigsvnjSUB34otIrUbOVAJ+4In+es28GvCsxvnjSVBMfNBGwb4qNUc2xnbPvALSXGF88aaiZb6Y1BX5HK0kvs2YUOsHfCN1F1PjiqYDoJBc0+ucz8Jm/vwfTMccXTyH6qvRea/kI/vGJHkX4r2588RQi3en9B+ACLKloj17K8Ra8j56KVpJPbFL5Ix893aiDjTWIJzc0IJ7i0TVZfVwUyNT7qAAAAP50RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+KDwvbW8+PG1mcmFjPjxtbj41PC9tbj48bXJvdz48bW4+MjwvbW4+PG1zcXJ0Pjxtbj4yPC9tbj48L21zcXJ0PjwvbXJvdz48L21mcmFjPjxtbz4sPC9tbz48bW8+JiN4QTA7PC9tbz48bWZyYWM+PG1uPjU8L21uPjxtcm93Pjxtbj4yPC9tbj48bXNxcnQ+PG1uPjI8L21uPjwvbXNxcnQ+PC9tcm93PjwvbWZyYWM+PG1vPik8L21vPjwvbWF0aD4SASIDAAAAAElFTkSuQmCC)
Slope of tangent =
= ![fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADoAAAAxCAYAAACLfLrrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAeOiuv/QAAAulJREFUeNrtmU+EVVEcx6+h0Z8RNcbQ7DKLFhlD8iSVSMbzekabjEqeaFFkdikyi0S1aJEoMpIkxmiRJDKSSpskSYuYNk9tIpnFSJvX96fv5Tjuufe8e+/M/fWcLx/evefd455/v3Pu7xtF1WtdpFzLoK9gHVtAW3NjR8HHEuq5Da5oHs1J8LCEzvoBNmlp1AC4AX6B32Ae3ATnWD4LjiQ8Nw0up9QrHXVeSyNlDb4ALf7eCGZAh6MqOgGuWc+tB1/AoKPecfCN/1Ohi8bImZJGjPB3HcxZ5TN81qWn4LSmtdjm1LVHedm4llH7ZFwPgUWw1lHnXpavKRjhd4NHnBkvQS1vI8dZga0ap7P9IrGuc3269AYcLRjha2xkn9W5uXQY3E+4PwXuWvdega1kMWV/PcSX7mb/TYrwH4ylE+sZGMu7hcwl3JfGn7Lu3QNN8AAcTwls0shGgQgvOpDQ0XEUr+dpqETEz5zCoh3gCddEPWErmc04RBwDrwtGeNEdzjZbMpUP5p2+w6xAevct2A9+JgSaSb5Q01GPBJ6vYE/BCB+x8zsOhlc6OjfZES6d4ZZSNMLL9ZJjNiytxjYkjdjlKNsAvhtLoEiElxFbSPjfTq7nFdV2rl2XLnici30jfLytRAkHlFaVB47NPLiPlhThB7i1mJKPgvegv8qGXgW3So7wcgo7y9/bGBsaVWcO2vy4LjPCj3FU/3A7a2o4K6tPkwQFBelTp0cJCgoK+v9PUGUYTLHUGk1lGUyx1BpNZRhMZqepMJp80o8nHSNyiWVpUmE0+aYfI37hmxm4FjskTWqMJt/0o2gi+mdFiCTBvOBRvxqjySf9aOo52Mev/8GMutUYTd0YTLGmmd7w8T/UGE3dGExxz85z+k5l1K3KaOrGYBIH7TGDikz1dylTV53R5Jt+HGJQMRvWcMwGkUqjKSv92M/yEUfPTlj3esZoylLPGE1p6hmjKUvBaMoR6YPRVHXmIBhNq6m/l2UoLP+SwzIAAAFmdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtc3R5bGUgZGlzcGxheXN0eWxlPSJ0cnVlIj48bWZyYWMgYmV2ZWxsZWQ9InRydWUiPjxtcm93PjxtaT5kPC9taT48bWk+eTwvbWk+PC9tcm93Pjxtcm93PjxtaT5kPC9taT48bWk+JiN4M0I4OzwvbWk+PC9tcm93PjwvbWZyYWM+PC9tc3R5bGU+PG1zdHlsZSBkaXNwbGF5c3R5bGU9InRydWUiPjxtZnJhYyBiZXZlbGxlZD0idHJ1ZSI+PG1yb3c+PG1pPmQ8L21pPjxtaT54PC9taT48L21yb3c+PG1yb3c+PG1pPmQ8L21pPjxtaT4mI3gzQjg7PC9taT48L21yb3c+PC9tZnJhYz48L21zdHlsZT48L21mcmFjPjwvbWF0aD74luT2AAAAAElFTkSuQmCC)
![fraction numerator d y over denominator d theta end fraction space equals space 3 cos theta space minus space 3 sin squared theta cos theta
fraction numerator d x over denominator d theta end fraction space equals space minus 3 sin theta space plus thin space space 3 cos squared theta sin theta](data:image/png;base64,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)
= ![fraction numerator begin display style bevelled fraction numerator d y over denominator d theta end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d theta end fraction end style end fraction space equals space fraction numerator 3 cos theta space minus space 3 sin squared theta cos theta over denominator negative 3 sin theta space plus space 3 cos squared theta sin theta end fraction](data:image/png;base64,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)
Slope of tangent at point P(
= ![fraction numerator 3 cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style space minus space 3 cross times begin display style 1 half end style cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator negative 3 cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style space plus space 3 cross times begin display style 1 half end style space cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction space equals space fraction numerator begin display style bevelled fraction numerator 3 over denominator 2 square root of 2 end fraction end style over denominator begin display style bevelled fraction numerator negative 3 over denominator 2 square root of 2 end fraction end style end fraction space equals space minus 1](data:image/png;base64,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)
When slope of tangent at P= -1
Slope of normal at point P = 1
Equation of normal at point P
y - y1 = m(x-x1)
![y space minus space fraction numerator 5 over denominator 2 square root of 2 end fraction space equals space 1 open parentheses x minus space fraction numerator 5 over denominator 2 square root of 2 end fraction close parentheses space rightwards double arrow space y space equals space x
W h e n space y space equals space x comma space t h i s space m e a n s space t h e space l i n e space p a s s e s space t h r o u g h space t h e space o r i g i n space left parenthesis 0 comma 0 right parenthesis](data:image/png;base64,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)
Related Questions to study
Maths-
The normal of the curve given by the equation x = a (sin
+ cos
), y = a (sin
– cos
) at the point Q is -
The normal of the curve given by the equation x = a (sin
+ cos
), y = a (sin
– cos
) at the point Q is -
Maths-General
Maths-
If the tangent at ‘t’ on the curve y =
, x =
meets the curve again at
and is normal to the curve at that point, then value of t must be -
If the tangent at ‘t’ on the curve y =
, x =
meets the curve again at
and is normal to the curve at that point, then value of t must be -
Maths-General
Maths-
The tangent at (
,
–
) on the curve y =
–
meets the curve again at Q, then abscissa of Q must be -
The tangent at (
,
–
) on the curve y =
–
meets the curve again at Q, then abscissa of Q must be -
Maths-General
Maths-
If
= 1 is a tangent to the curve x = 4t,y =
, t
R then -
If
= 1 is a tangent to the curve x = 4t,y =
, t
R then -
Maths-General
Maths-
The line
+
= 1 touches the curve
at the point :
The line
+
= 1 touches the curve
at the point :
Maths-General
physics
An object moves in a straight line. It starts from the rest and its acceleration is . 2 ms2.After reaching a certain point it comes back to the original point. In this movement its acceleration is -3 ms2, till it comes to rest. The total time taken for the movement is 5 second. Calculate the maximum velocity.
An object moves in a straight line. It starts from the rest and its acceleration is . 2 ms2.After reaching a certain point it comes back to the original point. In this movement its acceleration is -3 ms2, till it comes to rest. The total time taken for the movement is 5 second. Calculate the maximum velocity.
physicsGeneral
Maths-
For the ellipse
the foci are
For the ellipse
the foci are
Maths-General
Maths-
For the ellipse
, the latus rectum is
For the ellipse
, the latus rectum is
Maths-General
Maths-
The sum of distances of any point on the ellipse 3 x2 + 4y2 = 24 from its foci is
The sum of distances of any point on the ellipse 3 x2 + 4y2 = 24 from its foci is
Maths-General
Maths-
The equations x = a
represent
The equations x = a
represent
Maths-General
Maths-
The equations x = a cos q, y = b sin q, 0 ≤ q < 2 p, a ≠ b, represent
The equations x = a cos q, y = b sin q, 0 ≤ q < 2 p, a ≠ b, represent
Maths-General
Maths-
The line y = 2x + c touches the ellipse
if c is equal to
The line y = 2x + c touches the ellipse
if c is equal to
Maths-General
Maths-
The eccentricity of the conic 3x2 + 4y2 = 24 is
The eccentricity of the conic 3x2 + 4y2 = 24 is
Maths-General
Maths-
The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity
is
The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity
is
Maths-General
Maths-
The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if
The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if
Maths-General