Q: Let R ^2 denote R R . Let S= \ (a, b, c): a, b, c R and a x^2+2 b x y+c y^2>0 for all (x, y) R ^2-\ (0,0)\ \ . Then which of the following statements is (are) TRUE?

$ aligned & a x^2+2 b x y+c y^2>0 \\ & y, x R -\ (0,0)\ \\ & c ( y x )^2+2 b ( y x )+a>0 \\ & 4 b^2-4 a c 2 6$  option A is incorrect $ aligned & (B) if (3, b, 1 12 ) S \\ & b^2 0 aligned $  unique solution  option D is correct.

Question 1

Let R ^2 denote R R . Let S= \ (a, b, c): a, b, c R and a x^2+2 b x y+c y^2>0 for all (x, y) R ^2-\ (0,0)\ \ . Then which of the following statements is (are) TRUE?

Accepted Answer

$ aligned & a x^2+2 b x y+c y^2>0 \ & y, x R -\ (0,0)\ \ & c ( y x )^2+2 b ( y x )+a>0 \ & 4 b^2-4 a c 2 6$  option A is incorrect $ aligned & (B) if (3, b, 1 12 ) S \ & b^2 0 aligned $  unique solution  option D is correct.

Question 2

Let P Q R S be a quadrilateral in a plane, where Q R = 1 , &#8736; P Q R = &#8736; Q R S = 70 &#176; , &#8736; P Q S = 15 &#176; and &#8736; P R S = 40 &#176; . If &#8736; R P S = &#952; &#176; , P Q

Accepted Answer

Given, P Q R S be a quadrilateral in a plane, where Q R = 1 , &#8736; P Q R = &#8736; Q R S = 70 &#176; , &#8736; P Q S = 15 &#176; and &#8736; P R S = 40 &#176; . If &#8736; R P S = &#952; &#176; , P Q = &#945; and P S = &#946; ,  Now plotting the diagram of given values we get,   Applying sine rule in &#8710; P Q R  we get, &#945; sin 30 &#176; = 1 sin 80 &#176; &#8658; &#945; = 1 2 sin 80 &#176; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; . . . i Applying sine rule

Question 3

Consider a triangle &#916; whose two sides lie on the x -axis and the line x + y + 1 = 0 . If the orthocenter of &#916; is  ( 1 , 1 ) , then the equation of the circle passing through the vertice

Accepted Answer

Let  A  is the point of intersection of the given lines  y = 0 , &#160; x + y + 1 = 0 Their point of intersection  A  is  - 1 , 0 Let  B  is the other point on the line  x + y + 1 = 0 , Now, the perpendicular from  B  on the x -axis will pass through the orthocenter. So, the equation of perpendicular from  B  on the x -axis is  x = 1 Thus, the coordinates of  B  are  1 , - 2 Let the third vertex is  h

Question 4

Consider the lines L_ 1 and L_ 2 defined by L_ 1 : x 2 +y-1=0 and L_ 2 : x 2 -y+1=0 For a fixed constant , let C be the locus of a point P such that the product of the distance of P from L_ 1 and the

Accepted Answer

Let the point  P  is  h , k Distance of  P  from  L 1 = h 2 + k - 1 ( 2 ) 2 + 1 2 = h 2 + k - 1 3 Distance of P from L 2 = h 2 - k + 1 ( 2 ) 2 + 1 2 = h 2 - k + 1 3 &#8756;  The equation of the locus of P is h 2 + k - 1 3 &#215; h 2 - k + 1 3 = &#955; 2 h 2 + k - 1 3 h 2 - k + 1 3 = &#955; 2 &#8658; 2 h 2 - ( k - 1 ) 2 = 3 &#955; 2 Hence, the equation of the locus is  2 x 2 - ( y - 1 ) 2 = 3 &#955; 2 The line is  y = 2 x + 1  or  y - 1

Question 5

Consider the lines L_ 1 and L_ 2 defined by L_ 1 : x 2 +y-1=0 and L_ 2 : x 2 -y+1=0 For a fixed constant , let C be the locus of a point P such that the product of the distance of P from L_ 1 and the

Accepted Answer

From the first question The equation of the locus is  2 x 2 - ( y - 1 ) 2 = 27 The line is  y = 2 x + 1  or  y - 1 = 2 x By substituting the value of y  in the equation of the curve  C , we get 2 x 2 - ( y - 1 ) 2 = 27 &#8658; 2 x 2 - ( 2 x ) 2 = 27 &#8658; &#160;&#160; 2 x 2 = 27 &#8658; &#160;&#160; x = &#177; 3 3 2 &#8658; x 1 , &#160; x 2 = &#177; 3 3 2 Let M  be the mid-point of  R ' &#160; &#38; &#160; S ' So, the  x  coordinat

Question 6

Let O be the center of the circle x 2 + y 2 = r 2 ,  where  r &#62; 5 2 . Suppose P Q  is a chord of this circle and the equation of the line passing through P  and Q  is&nbsp

Accepted Answer

Circumcenter C of Δ O P Q lies on x + 2 y = 4 Let C ≡ 4 - 2 α , α ∵ O C ⊥ P Q ⇒ s l o p e o f O C · s l o p e o f P Q = - 1 ⇒ α 4 - 2 α - 2 4 = - 1 ⇒ α = 8 5 ⇒ C 4 5 , 8 5 O M = | 0 + 0 - 5 | 2 2 + 4 2 = 5 2 C M = 2 4 5 + 4 8 5 - 5 2 2 + 4 2 = 3 2 5 O C = P C = 4 5 2 + 8 5 2 = 4 5 Now, P M 2 = O P 2 - O M 2 = P C 2 - C M 2 ⇒ r 2 - 5 4 = 16 5 - 9 20 ⇒ r = 2

Question 7

Let the point B be the reflection of the point A 2 , &#160; 3 with respect to the line 8 x - 6 y - 23 = 0 . Let T A and T B be circle of radii 2 and 1 with centres A and B respectively. Let T be a com

Accepted Answer

Clearly, A C P and B C Q are similar triangles. Hence, A C B C = r 1 r 2 A C B C = 2 A C = 2 B C ...(i) Now, since B , is image of A in line 8 x - 6 y = 23 A D = B D A B = 2 A D ...(ii) A D = distance of A from line 8 x - 6 y = 23 A D = 16 - 16 - 23 10 = 5 2 Now as A C = 2 B C ⇒ A C = 2 ( A C - A B ) ⇒ A C = 2 A B ⇒ A C = 4 A D (from (ii)) A C = 4 × 5 2 A C = 10

Straight Lines — Previous Year Questions & Solutions