Question

P(2, 2) and $Q(\frac{1}{2},-1)$ are two points on the parabola $y^2=2x$. The coordinates of the point R on the parabola, where the tangent to the curve is parallel to the chord PQ, is

• A. $(\frac{5}{4},\sqrt{\frac{5}{2}})$
• B. $(2,-1)$
• C. $(\frac{1}{8},\frac{1}{2})$
• D. None of these

Answer 1

The correct option is C $(\frac{1}{8},\frac{1}{2})$

Tangent to $y^2=2x$ at point $(x_1,y_1)$ is
$y{y}_1=(x+x_1)$
Slope of $PQ=\frac{-1-2}{\frac{1}{2}-2}$
= $\frac{-3}{-3/2}$
= 2
$y=\frac{1}{y_1}(x+x_1)$
$\frac{1}{y_1}=2(y_1=1/2)$
$y_1^2={2x}_1$
$\therefore x_1=1/8$
$\therefore$ point is $(\frac{1}{8},\frac{1}{2})$