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ICSE Class 10 Answered

A hemisphere and a cone have equal bases.If their heights are also equal,then what is the ratio of their curved surfaces. 2.find the image of the point (-3,1) in the line 2x-3y=4.
Asked by Farheen1447 | 07 Mar, 2019, 06:10: PM
answered-by-expert Expert Answer
curved surface of hemisphere = 2πr2 sq.units
 
curved surface of cone = begin mathsize 12px style straight pi space straight r space square root of straight r squared plus straight r squared end root space equals space square root of 2 space straight pi space straight r squared end style sq.units  
[ radius of base of hemisphere and base of cone are same. Height of the hemisphere is same as radius.
Height of cone is same as height of hemisphere ]
 
ratio of curved surface area of hemisphere to that of cone = √2
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point P(-3, 1) is on the line which is perpendicular to the line 2x-3y = 4.  
 
Hence point P(-3, 1) is on the line 3x+2y = k  , where k is to be determined .
 
3(-3) + 2(1) = - 7    Hence the perpendicular line is 3x+2y = -7
 
Coordinates of point of intersection Q is obtained by solving the equations of the two lines
 
2x -3y  = 4    ........................(1)
3x +2y = -7   ......................(2)
 
eqn.(1)×3   is :  6x  -9y  = 12   ................(3)
eqn.(2)×2   is :  6x +4y  = -14   ...............(4)
 
eqn.(3) - eqn.(4) gives,  -13y = 26   or  y = -2
 
substituting y = -2 in eqn.(1),  we get x = -1
 
hence point of intersection is Q (-1, -2)
 
Q is the middle point of P and image of P i.e., P'(x,y)
 
hence (x-3)/2 = -1   or  x = 1  ;  similarly  (y+1)/2  = -2  or  y = -5
 
hence image point  is ( 1, -5)
Answered by Thiyagarajan K | 07 Mar, 2019, 08:34: PM

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