Mathematics - Straight Lines Question with Solution | TestHub
Match all applicable values. For (D), each listed λ needs some real line pair of the stated kind.
List I | List II |
(A) k if 4x²+8xy+ky²=9 represents a pair of straight lines. | (P) 3 |
(B) C if slopes of x²-2Cxy-7y²=0 have sum four times their product. | (Q) -3 |
(C) h if one line of x²+hxy+2y²=0 has twice the slope of the other. | (R) 2 |
(D) λ for which real pairs ax²+2hxy+by²=0 and ax²+2hxy+by²+λ(x²+y²)=0 are equally inclined. | (S) 4 |
Options:
Answer:
Solution:
(A) The quadratic must be a perfect square: , so . (B) With slopes , their sum is and product , whence . (C) If the slopes are , their product is ; hence and their sum , giving . (D) Adding preserves the traceless part of the associated quadratic form and hence its angle-bisector directions. Any of the four listed can occur for some sufficiently separated real pair of lines, for example take before the shift.
