Find the value a and b such that the cone ax^2 + by^2 + z^2 = 0 and the ellips 2x^2 + 4y^2 + y^2 = 42 intersect and are perpendicular to each other at the point (1, 3, 2)

ruigE 2021-02-26 Answered

Find the value a and b such that the cone ax2+by2+z2=0 and the ellips 2x2+4y2+y2=42 intersect and are perpendicular to each other at the point (1, 3, 2)

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Expert Answer

Benedict
Answered 2021-02-27 Author has 108 answers

Step 1 We are given that the cone ax2+by2+z2=0 and the ellipsoid 2x2+4y2+z2=42 intersect and are perpendicular to each other at the point (1, 3, 2). So, both the equation intersect. Thus, ax2+by2+z2=2x2+4y2+z242
ax22x2+by24y2+z2z2+42=0
(a2)x2+(b4)y2+42=0 ........(1) So, the locus of intersection of the cone and the ellipsoid is the circle written above. We further are given that both the conic sections are perpendicular to each other at the point (1, 3, 2). So, the equation (1) must be passing through the point (1, 3). Thus, (a2)x2+(b4)y2+42=0 (a2)+4(b4)+42=0 a+4b+24=0 ........(2) Step 3 Direction ratio of a function F(x,y,z)=0 is nothing but the gradient of F. So, the direction ratios of ax2+by2+z2=0is<2ax,2by,2z>. Then the direction ratios at (1,3,2)is<2a,6b,4>. Now, the direction ratios of 2x2+4y2+z242=0is<4x,8y,2z>. Then the direction ratios at (1,3,2)is<4,24,4>. We have that the conic sections are perpendicular to each other at the point (1, 3, 2). So, 8a+144b+16=0a+18b+2=0 .......(3) Step 4 Now, solving equation (2) and (3) to find the value of a and b. a+18b+2=0
a+4b+24=0 Isolate a for a+18b+2=0:a=18b2 Subtitute a=18b2 18b2+4b+24=0 Isolate bfor18b2+4b+24=0:b=117 For a=18b2 Subsititute b=117 The solution to the system of equations are: b=117,a=2127 Step 5 Thus, the equation of the cone is, 2127x2+117y2+z2=0

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