Question

Consider the parabola $x^2+4y^2-4xy-10x+5=0$. Which of the following statements are true for above parabola?

• A. Equation of tangent at vertex of parabola is $2x+y-1=0$
• B. Equation of tangent at vertex of parabola is $8x+4y-5=0$
• C. Equation of axis of parabola is $x-2y-1=0$
• D. Latus rectum of parabola is $\frac{4}{\sqrt{5}}$

Answer 1

Answer: (A), (C), (D)

The correct options are
A Equation of tangent at vertex of parabola is $2x+y-1=0$
C Latus rectum of parabola is $\frac{4}{\sqrt{5}}$
D Equation of axis of parabola is $x-2y-1=0$

Given equation of parabola

$P:x^2+4y^2-4xy-10x+5=0$
It can be represented as
$(x-2y-1{)}^2=4(2x+y-1)$
$\Rightarrow {\left[\frac{x-2y-1}{\sqrt{5}}\right]}^2=\frac{4}{\sqrt{5}}\left[\frac{2x+y-1}{\sqrt{5}}\right]$

Comparing it with $Y^2=4aX$, we get
Axis $Y=0$ is same as $x-2y-1=0$
Tangent at vertex $X=0$ is same as $2x+y-1=0$
Latus rectum $4a$ is same as $\frac{4}{\sqrt{5}}$

Hence, option A, C and D.