If values of a, for which the line `y=ax+2sqrt(5)` touches the hyperbola `16x^2-9y^2 = 144` are the roots of the equation `x^2-(a_1+b_1)x-4=0`, then the values of `a_1+b_1` is
A. 2
B. 4
C. zero
D. none of these
1 Answers
Talk Doctor Online in Bissoy App
If `S=[((sqrt(3)-1)/(2sqrt(2)),(sqrt(3)+1)/(2sqrt(2))),(-((sqrt(3)+1)/(2sqrt(2))),(sqrt(3)-1)/(2sqrt(2)))], A=[(1,0),(-1,1)]` and `P=S ("adj.A") S^(T)
`S=[((sqrt(3)-1)/(2sqrt(2)),(sqrt(3)+1)/(2 sqrt(2))),(-((sqrt(3)+1)/(2sqrt(2))),(sqrt(3)-1)/(2sqrt(2)))]` `=[("sin "15^(@),cos 15^(@)),(-"cos "15^(@),sin 15^(@))]` `:. SS^(T)=S^(T)S=I` Now, `S^(T) P^(10) S=S^(T)(S ("adj. A")S^(T))^(10)S` `=S^(T)S("adj. A") S^(T) (S("adj. A")S^(T))^(9)S` `=I ("adj. A")S^(T) (S("adj. A")S^(T))^(9)S` `=("adj. A")S^(T)S("adj. A") S^(T) (S("adj....
2 Answers 1 views
A variable chord of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,(b > a),` subtends a right angle at the center of the hyperbola if this chord touches. a
Correct Answer - A Let the variable chord be `x cos alpha+y sin alpha=p" (1)"` Let this chord intersect the hyperbola at A and B. Then the combined equation of OA...
2 Answers 1 views
The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consid
Correct Answer - D Centre `-=(1,2)` Radius of auxiliary circle `=a =sqrt((2-1)^(2)+(5-2)^(2))=sqrt(10)` `2ae=sqrt(8^(2)+8^(2))=8sqrt2 or e=(4)/(sqrt5)` `b^(2)=a^(2)e^(2)-a^(2)=32-10=22` `"or "2b=2sqrt(22)`
2 Answers 2 views
If `0<a_1<a_2<ddot<a_n ,` then prove that `tan^(-1)((a_1x-y)/(x+a_1y))+tan^(-1)((a_2-a_1)/(1+a_2a+1))+tan^(-1)((a_3-a_2)/(1+a_3a_2))++tan^(-1)((a_n-a_
Here, `tan^(_1) ((a_(1) x- y)/(a + a_(1) y)) = tan^(-1) ((a_(1) - (y)/(x))/(1 + a_(1) (y)/(x))) = tan^(-1) a_(1) - tan^(-1) (y)/(x)` `tan^(-1) ((a_(2) -a_(1))/(1 + a_(2) a_(1))) = tan^(-1)...
2 Answers 1 views
If `|x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1|` then the two triangles with vertices `(x_1, y_1),(x_2,y_2),(x_3,y_3)` and `(a_1,b_1),(a_2,b
Correct Answer - A We have `(1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|or (1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=(1)/(2)|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|` Hence, the area of triangle with vertices `(x_1,y_1),(x_2,y_2),(x_3,x_3)` is the same as the area of triangle with vertices `(a_1,b_1),(a_2,b_2),(a_3,b_3)`. Hence, the two triangles...
2 Answers 1 views
If the equation of the locus of a point equidistant from the points `(a_1,b_1)` and `(a_2,b_2)` is `(a_1-a_2)x+(b_2+b_2)y+c=0`, then the value of C is
Correct Answer - D Let (h,k) be the point on the locus. Then by the given conditions, `(h-a_1)^1+(k-b_1)^2=(h-a_2)^2+(k-b_2)^2` or `2h(a_1-a_2)+2k(b_1-b_2)+a_2^2+b_2^2-a_1^2=0` `h(a_1-a_2 )+k(b_1-b_2)+(1)/(2)(a_2^2+b_2^2-a_1^2-b_1^2)=0` .....(1) Also, since (h,k) lies on the given locus,...
2 Answers 1 views
If `p` is the probability that a man aged `x` will die in a year, then the probability that out of `n` men `A_1,A_2, A_n` each aged `x ,A_1` will die
Correct Answer - C Let the probability that a man aged x dies in a year p. Thus the probability that a man aged x does not die in a year...
2 Answers 1 views
If the tangent at (1,7) to curve `x^(2)=y-6` touches the circle `x^(2)+y^(2)+16x+12y+c=0` then the value of c is
Correct Answer - A 1 Differentiating curve `x^(2)=y-6` w.r.t. x, we get `(dy)/(dx)=2x` Therefore, slope of tangent to the curve at point P(1,7) is `((dy)/(dx))_((1","7))=2` Equation of tangent to the curve...
2 Answers 2 views
In ` A B C ,s i d e sb , c` and angle `B` are given such that `a` has two valus `a_1a n da_2dot` Then prove that `|a_1-a_2|=2sqrt(b^2-c^2sin^2B)`
`cos B = (c^(2) + a^(2) - b^(2))/(2ca)` or `a^(2) - (2c cos B) a + c^(2) - b^(2) = 0` This equation has roots `a_(1) and a_(2)` `rArr a_(1)...
2 Answers 1 views