Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
MATCH THE FOLLOWING:
(A) The number of integral values of $u$ for which point $(a, a^2)$ lies completely inside the triangle formed by lines $x = 0, y = 0.2y + x = 3$
(B) A point on the line $x + y = 4$ which lies at a unit distance from the line $4x + 3y = 10$ has the coordinates $(a, \beta)$ then possible value of $a + \beta$
(C) If $(a, \beta)$ be the Orthocenter of triangle made by lines $x + y = 1, x - y + 3 = 0, 2x + y = 7$ then the value of $a + \beta$ is
(D) In a triangle $ABC$, the bisector of angles $B$ and $C$ lie along the lines $y = x$ and $y = 0$. If $A$ is $(1, 2)$ then $\sqrt{10} d(A, BC)$ equals (where $d(A, BC)$ denotes the perpendicular distance of $A$ from $BC$.)
Step-by-Step Solution
Key Concept: Factor quadratic inequalities carefully and use the perpendicular distance formula to find parameter values.
(A) For $2a^2 + a - 3 0$, we have $a \in (0, 1)$, so there are no integral values of $a$. (B) The perpendicular distance from $P(4, \lambda)$ to the line $4x + 3y = 10$ is $\frac{|4(4) + 3(\lambda) - 10|}{5} = 1$, which gives $|4\lambda + 6| = 5$, so $\lambda = -\frac{1}{4}$ or $\lambda = -\frac{11}{4}$. The answer options are $[A-p]$, $[B-s]$, $[C-q]$, $[D-s]$.
Correct Answer: [A-p] [B-s] [C-q] [D-s]