Circles
Circle Through Limit of Intersection of Lines
nta_pyq_2024_apr
Grade 11
Question:
Let a circle passing through $(2,0)$ have its centre at the point $(h,k)$. Let $(x_c,y_c)$ be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5c^2y=1$. If $h=\lim_{c\to1}x_c$ and $k=\lim_{c\to1}y_c$, then the equation of the circle is:
$25x^2+25y^2-2x+2y-60=0$
$5x^2+5y^2-4x+2y-12=0$
$5x^2+5y^2-4x-2y-12=0$
$25x^2+25y^2-20x+2y-60=0$
Step-by-Step Solution
Key Concept: Solve the two lines for $x_c,y_c$, then take $c\to1$. $h=\lim_{c\to1}x_c=2/5$, $k=\lim_{c\to1}y_c=-1/25$. Circle passes through $(2,0)$ with centre $(2/5,-1/25)$.
Centre $(2/5,-1/25)$. Circle $(x-2/5)^2+(y+1/25)^2=r^2$, passing through $(2,0)$: $r^2=161/125$... after clearing: $25x^2+25y^2-20x+2y-60=0$.
Correct Answer: 4