Question

Let P be the point (1, 0) and Q a point of the locus $y^{2} = 8x$. The locus of mid point of PQ is.

• A. $x^{2} + 4y + 2 = 0$
• B. $x ^ { 2 } - 4 y + 2 = 0$
• C. $y ^ { 2 } - 4 x + 2 = 0$
• D. $y ^ { 2 } + 4 x + 2 = 0$

Answer 1

Answer: (C)

$y ^ { 2 } - 4 x + 2 = 0$

Answer 2

$P = ( 1,0 ) , Q = ( h , k )$ such that $k ^ { 2 } = 8 h$

Let $( \alpha , \beta )$ be the midpoint of $P Q$ ;

$\alpha = \frac { h + 1 } { 2 } , \beta = \frac { k + 0 } { 2 } ; 2 \alpha - 1 = h , 2 \beta = k$

$( 2 \beta ) ^ { 2 } = 8 ( 2 \alpha - 1 ) \Rightarrow \beta ^ { 2 } = 4 \alpha - 2 \Rightarrow y ^ { 2 } - 4 x + 2 = 0$.