Parabola & Ellipse
Common tangent to circle and ellipse — normal to parabola
MJAT_TS8_P1
Grade 12
Question:
The common tangent to the circle $x^2+y^2=16$ and ellipse $\dfrac{x^2}{64}+\dfrac{y^2}{4}=1$ in the form $y=mx+c$ is also a normal to the parabola $y^2=4ax$. If normal at $A(x_1,y_1)$ meets the parabola again at $B(x_2,y_2)$, the value of $95(x_1+y_2)$ is:
Step-by-Step Solution
Key Concept: Common tangent to circle $r=4$ and ellipse: $\frac{4}{\sqrt{1+m^2}}=\sqrt{64m^2+4}\Rightarrow 16(1+m^2)=64m^2+4\Rightarrow 12=48m^2\Rightarrow m=-1/2$. Normal to $y^2=4ax$ with slope $m=-1/2$: $y=-\frac{1}{2}x+2a\cdot\frac{1}{2}+\frac{a}{8}\cdot(-1)^3\cdot 4$.
$95(x_1+y_2)=\mathbf{148}$.
Correct Answer: 148